a priori estimates for solutions to elliptic equations on non-smooth domains

Proceedings of the Royal Society of Edinburgh, 132A, 793-813, 2002

A priori estimates for solutions to elliptic equations on non-smooth domains

Daniel Daners School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia (D. Daners(Qmaths. usyd. edu. au)

(MS received 7 September 2000; accepted 6 August 2001)

It is proved that elliptic boundary-value problems have a global smoothing property in Lebesgue spaces, provided the underlying space of weak solutions admits a Sobolev-type inequality. The results apply to all standard boundary conditions, and a wide range of non-smooth domains, even if the clas!3ical estimates fail. The dependence on the data is explicit. In particular, thi~ provides good control over the domain dependence, which is important for applications involving varying domains.

1. Introduction

If D C JRN is a bounded domain of class C2 , then it is well known that every solution to the elliptic boundary-value problem . ,., .

-~u=f in D, }

on aD (1.1) au

av + f3u = 0

lies in the Sobolev space W;(D) if f E Lp(D) and p E (1, (0). Here, v is the outer unit normal to aD and f3 E [0,00]' with f3 = 00 corresponding to first (Dirichlet) boundary conditions. By the well-known W;-regularity theory for elliptic equations on smooth domains (see [2, ch. V]), there exists a constant c > 0 such that Ilullw; :::; c(Uftlp + Ilullp) for all solutions u of (1.1) with f E Lp(fl). The Sobolev embedding theorem (see [1, ch. 5]) asserts that W;(D) Y Lm(p) (D), with m(p) := Np/(N - 2p) if p E (1, ~N) and m(p) = 00 if p > ~N. Hence there exists a constant c > 0 such that

(1.2)

for every solution of (1.1) with f E Lp(D). If fl is just Lipschitz, then it is also well known that f E Lp(D) does not imply that every solution of (1.1) li~s in W;(flr For an extensive discussion of this problem, we refer the reader to [12,14,18].

The: purpose of this paper is to discuss under what conditions a global estimate of the form (1.2) remains valid if D is non-smooth. Throug4~ut we will rep~ace th~, ~aplace operator with a uniformly elliptic operator in divergence form with r~al bounded and measurable coefficients, not necessarily sel£':'adjoint. The basic r~s~lt is the following. Suppose that V c wi (fl) is the Hilb~rt space appearing

I

© 2002 The Royal Society of Edinburgh

793

794 D. Daners

in the weak setting of problem (1.1). Then every solution of (1.1) with f E Lp(fl) satisfies (1.2), provided V admits a Sobolev inequality, that is,

(1.3)

More generally, we will show that such an Lp-Lq-smoothing property of (1.1) remains true if V Y Lr(fl) for some r > 2. Setting d .- 2rj(r - 2), we can rewrite the embedding as

(1.4)

and we will be able to prove an a priori estimate for solutions to (1.1) similar to (1.2), with N substituted by some d > max{2, N}. This means that (1.1) has a smoothing property in Lebesgue spaces even if the classical Wi-estimates fail. Moreover, if d = N, the exponent m(p) is exactly the same as the one we get from the embedding of W;(fl), even if the solution is not in that space. The embedding (1.3) holds always for the first boundary-value problem. It also holds for a wide range of non-smooth domains for other boundary conditions, including some domains with a fractal boundary. Domains satisfying (1.4) for d > N include domains with outward pointing polynomial cusps. For more details, we refer to the examples given in § 3.

One important feature of our approach is that we get 'explicit' estimates.: In particular, we get good control over the constant c in (1.2) in terms of the embedding constant of (1.4). The embedding constant is a property of the domain and the boundary conditions only. If we have control over this constant, we get control over an Lq-norm of the solution in terms of the Lp-norm of the right-hand side for some q > p, uniformly with respect to the domains. This is particularly useful when dealing with problems on varying domains, even when only looking at smooth domains (see, for instance, [5,7,8]).

We further show that, for self-adjoint problems on bounded domains, the converse is also true, namely that (1.2) for some m(p) > p implies (1.4). Fi~ally, we will show that (1.1) has compact resolvent if (1.4) holds for some d > 2.

The idea of the proof is based on an iteration method originally introduced by Moser [17]. We shall use an abstract version thereof applicable to most standard elliptic boundary-value problems. Almost all results in the literature concentrate on the case p > ~N, where local or global Lao-estimates are deduced (see, for instance, [11, ch. 8]). A discussion for p < ~N in the case of first boundary conditions can be found in [20, thm 4.2]. There are, however, sorrie additional structural assumptions we are able to remove. Similar results can be found in [4, appendix to ch. 3]. Moreover, the result is explicitly stated in [5, lemma 1]. For third (Robin) boundary conditions, domain-independent global Lp-Lq-bounds are established in [9]. Some aspects of the second (Neumann) boundary-value problem are discussed in [8]. There is no comprehensive treatment involving all standard boundary conditions. In this paper we shall present such a. treatment under one simple set of assumptions. It provides a much more explicit dependence on the data than most available results.

The paper is organized as follows. In § 2 we introduce the assumptions and present our main results. In § 3 we illustrate how to apply the results to some standard boundary-value problems. Section 4 deals with abstractielliptic problems. Here we implement an abstract version of the iteration method due to Moser [17]. In § 5 we

A priori estimates 795

apply the abstract results to the case of elliptic boundary-value problems, proving the main results in § 2. Finally, in § 6 we prove a converse to the regularity results, and in § 7 a compactness property of the resolvent.

2. Assumptions and main results

Throughout, we assume that D c JRN is an open set. Moreover, let A be an elliptic operator formally given by

Au = - t, 8i (~(ai,j8jU) + aiu) + t, bi8iu + COU' (2.1)

We assume that ai,j, ai, bi , Co E L(X)(D) for i = 1, ... , N, and that A is uniformly strongly elliptic, that is, there exists ao > 0 such that

N

Re L ai,j(x)~i~j ~ aol~12 (2.2) i,j=l

holds for all xED and ~ = (6, ... , ~N) E eN. Suppose that aD is the disjoint union of r l , r 2 and r 3 . (Note that r i (i = 1,2,3) are not necessarily assumed to be open and closed.) We consider the boundary operator B defined by

ulan on r l (1st or Dirichlet),

Bu:=

N N

~ (~( ai,Au) + aiu) Vi on r2 (2nd or Neumann), (2.3)

N N

~(~(ai,j8jU) + aiu) Vi + bou on T3 (3rd or Robin),

where v = (VI, , .. , VN) is the outward unit normal to aD and bo E L(X)(r3 ) is nonnegative. We study the regularity properties of weak solutions of the boundary-value problem

Au=j in D, } Bu = 0 on aD,

(2.4)

under various assumptions on the boundary of D and the operator B. To define what we mean by a weak solution of (2.4), we set

for all u, v E Wi(D), and

a(u,v) := ao(u, so) + r bou<pd1-lN~l' i r 3

(2.5)

whenever the expression is defined. Here, 1-lN-l is the (N -I)-dimensional Hausdorff measure on r3 . Note that this measure coincides with the ordinary surface measure

796 D. Daners

if r3 is Lipschitz (see [10, thm 3.2.3]). We call a(·, .) the form associated with (A, B). We next introduce the requirements on the space of test functions associated with (A, B). Before we do so, we define

O~(u) := {u E Ok(U) : suppu C U}

for all sets U C JRN and kEN U { 00 }, where supp u is the support of u. Further, let wi(n) be the usual Sobolev space of functions in L 2 (n) with derivatives in L 2 (n), and Wi(n) the closure of O~(n) in Wi(n).

ASSU~PTION 2.1. Let V be a Hilbert space satisfying

wi(n) y V Y wi(n),

and

{u E c~(ii \ Tt) : h, u2 d1lN_l < oo} C V.

Further, assume that the norm II·IIB given by

(2.6)

(2.7)

(2.8)

for all u E V is equivalent to the norm on V. Finally, for every piecewise smooth function 9 E O(JR) with g' E L=(JR) and g(O) = 0, we require that go u E V for all uE V.

As we assumed that bo ~ 0, the norm IluliB is well defined for all u E V.

DEFINITION 2.2. Suppose that V is a Hilbert space satisfying assumption 2.1 and that f E V'. We call u E V a weak solution of (2.4) in V if

a(u, v) = (f, v) (2.9)

is satisfied for all v E V, where

(f, v) := fa fv dx.

REMARK 2.3. Assume for the moment that the coefficients of (A, B) and n are smooth and that u E 0 1 (.0) n 02(Q) is a solution of (2.4). Then multiply the first equation in (2.4) by a function v E C~(tl \ r1 ). If we apply the divergence theorem, we see that u satisfies the integral identity (2.9), showing that u is a weak solution. On the other hand, every suffiCiently smooth weak solution of (2.4) turns out to be a classical solution. Condition' (2'.7) is to ensure that the boundary conditions of sufficiently smooth weak solutions hold pointwise.

REMARK 2.4. Note that, for some choices of n, the norm II·IIB on V is stronger than the one induced by Wi{n) (see [9, remark 3.5(£)]» Moreover, if r3 is very rough, it is not clear whether Irs u2 d1-lN-1 is finite for all u'E O~(tl \ r 1 ) (see [9, remark 3.5( d)]). .


Before we state our main result, let us introduce some quantities. We set

00 := Ila; bll~ + Ilco 1100' ao

(2.10)

where Co is the negative part of Co and

We further introduce a continuous seminorm qB on V by setting

qB(U) := (11\7ull~ + ao11Iu00IIL(r3

) + oBllull~)1/2 (2.11)

for all u E V, where OB = 0 or 1, depending on the particular boundary-value problem under consideration. Finally, we set

(2.12)

We are now in a position to formulate the main result of this section.

THEOREM 2.5. Let V be a Hilbert space satisfying assumption 2.1. Moreover, suppose that there exist constants d > 2 and CB > 0 such that

(2.13)

for all u E V. Finally, suppose that f E L2 (rl) n Lp (rl) and that u is a weak solution of (2·4) in V. Then u E Lp(rl), and there exists a constant C > 0 depending only on d and p such that

if p E [2, ~d) and

Ilull ~ {caolc~(llfllp + ABllull p) + Iluli p 00 ~ Caolc~lrll(2p-d)/dp(llfllp + ABllullp)

ifp > ~d.

if I rll is arbitrary,

if Irll < 00

(2.14)

(2.15)

Here, Irll denotes the Lebesgue measure of rl. For a modified problem, one can get rid of Ilulip on the right-hand side of (2.14) and (2.15), and still controlllulldp/(d_2p) and Ilull oo , respectively.

COROLLARY 2.6. Suppose that V satisfies assumption 2.1 and that f E L2(rl) n Lp(rl) for some p ~ 2. Moreover, assume that, for some d > 2 and CB > 0, the Sobolev-type inequality (2.13) holds. Then, for all A > AB, the problem

Au + AU = f in rl, } Bu = 0 on arl (2.16)

has a unique weak solution u E V . Moreover, there exists a constant c such that

798 D. Daners

where m(p) := dp/(d - 2p) if p E [2, ~d) and m(p) = 00 if P > ~d. Finally, the constant c only depends on d, p and upper bounds for ao1, CB and AB.

REMARK 2.7. If D is of finite measure, then one can estimate IIfl12 by IIfll p, More precisely, for A > AB, we have

IIullm(p) :::;;; c(l + (A - AB)-1) Ilfllp,

where c only depends on d, p and upper bounds for ao!' CB, AB and the measure of D. In particular, it follows that the resolvent operator of (2.16) is a bounded linear operator from Lp(D) to Lm(p) , with control over the norm if A > AB.

Based on the abstract regularity results in § 4, a proof of the above theorem will be given in § 5. The key ingredients are the Sobolev-type inequality (2.13), and an inequality generalizing the V-ellipticity of the form associated with (A,8) (see corollary 5.4). We only state our results in the case p ?: 2. Note, however, that the case p E (1,2) can be dealt with by duality. The idea is that the above estimates hold for the formal adjoint problem and p E (2, ~d). Hence the estimates follow by duality for p E (1,2) (for more details, compare [9, appendix]).

The inequality (2.13) implies that V Y L2d/(d-2)' If D has finite measure and A is large enough, then, by remark 2.7, we have Lp-Lq-estimates for weak solutions of (2.16) of the form

(2.17)

for some 2 ?: p < q ?: 00. In some cases, it turns out that the converse is also true.

THEOREM 2.8. Suppose that (A,8) is self-adjoint, that is, ai,j = aj,i and ai = bi

fOT' all i, j = 1, .. <., N. Further, assume that the measure of D is finite and that there exist constants C ?: 1, 2 :::;;; p < q :::;;; 00 and A E IR such that (2.17) holds for all solutions 0!(2.16). Then V Y L2d/(d-2) (D), with d:= 2pq(q - p)-1.

The proof of the above theorem will be given in § 6. Note that d is exactly what we expect from (2.14) by solving for d. As a consequence, d > 2 is optimal in (2.13) if and only if the 'exponent dp/(d - 2p) is optimal in theorem 2.5.

Another consequence of the smoothing properties is that, if D has finite measure, problem (2.16) has compact resolvent.

THEOREM 2.9. Suppose that D has finite measure and that the assumptions of theorem 2.5 hold. Then the resolvent of (2.16) is compact as an operator on Lp(D) for all p E [2,00).

The proof will be given in § 7. It follows from the observation that V is compactly embedded into L2(D) if an inequality of the form (2.13) holds.

3. Some examples of boundary-value problems

In this section we show how the results from § 2 apply to some standard boundaryvalue problems. In particular, we show how to get a Sobolev-type inequality of the form (2.13). It should be easy to apply theorem 2.5 to other cases. We start by showing that the space V satisfies assumption 2.1 for ~ll commonly considered boundary-value problems.


PROPOSITION 3.1. Suppose that r3 is Lipschitz open and closed in aD, and that ba E L oo(r3 ) is non-negative. We set

Va := {u E Wi(D) : suppu c .n \ rd

and define V to be the closure of Va in wi (D). Then V satisfies:assum:.ption 2.1.

Proof. Clearly, V satisfies (2.6) and (2.7). Moreover, it is obvious from the definitions that Ilullwi ~ IluliB for all u E V. As r3 is Lipschitz ope~ and closed, there exists a trace operator bounded from V into L 2(r3 ) (see [19]). Therefore, there exists C > 0 such that

for all u E V. Hence 11·llwi is equivalent to II·IIB on V. To verify the last assumption, let 9 E C(ffi.) be piecewise smooth with g' E Loo(ffi.) and g(O) = O. Let u E Va be arbitrary. Invoking the results in [11, § 7.4], it follows that 9 0 u E Wi(D). As g(O) = 0, we conclude that go u E Va. Finally, note that the map u M go u is continuous on Wi(D). Hence, by the density of Va in V, it' follows that go u E V for all u E V. Hence V satisfies assumption 2.1. D

3.1. The first boundary-value problem

In the first boundary-value problem, we have r 1 = aD. The space V to choose is V := Wi(D). By proposition 3.1, it satisfies assumption 2.1. We further set 8B = 0 and therefore

qB( u) = lI\7u112

for all u E V. By the standard Sobolev inequality (see [11, thm 7.10]), we have

for all u E Wi(D), where CN only depends on the dimension N. Hence theorem 2.5 applies, with CB = cN.If D has finite measure and N = 2, inequality (2.13) holds for all d >2 and the constant CB also depends on an upper bound for the measure of D. These facts are extensively used in [5].

3.2. The second boundary-value problem

In the second boundary-value problem we have r2 = aD. The space V to choose is V = Wi(D). By prop~sition 3.1 it satisfies assumption 2.1. We further set 8B := 1 and thus

qB(U) = IlullwiCn )

for all u E V. It is well known that an inequality of the form (2.13) is not true in general (see [1, thm 5.32]). If such an inequality is true, then the constant CB must depend on geometrical properties of D (see also [6, § 3]). Let us mention some classes of domains admitting an inequality of the form (2.13) for some d > 2. Note that, due to local regularity properties of functions in Wi(D), we always have d ?: N, so the best possible case is d = N.

800 D. Daners

(i) Domains satisfying an interior cone condition. If N ~ 3, we can choose d = N and the constant CB depends only on the cone, that is, its opening angle and its length. If N = 2, we can choose d E (2,00) arbitrary, but CB also depends on the measure of D. For a proof of these facts, we refer the reader to [1, lemmas 5.12 and 5.13]. In some applications, where the domain is varied, it is of interest to have uniform bounds on the Sobolev embedding constant CB. This is possible in some cases also without assuming a uniform interior cone condition. We refer to [8, § 2] for an example and some applications.

(ii) Extension domains, that is, domains for which there exists a bounded linear operator & : Wi(D) -+ Wi(IRN) such that &uln = u for all u E Wi(D). Lipschitz (or smoother) domains are extension domains (see [21, § VI.3.1]). However, there are much more irregular domains that are extension domains. In two dimensions, every quasi-disk is an extension domain (see [16, § 1.5.1], where also examples and characterizations of quasi-disks are given). Note that domains with a fractal boundary may be quasi-disks. More conditions for a domain to be an extension domain. are, for instance, given in [13].

(iii) There are classes of domains that satisfy a Sobolev-type inequality (2.13), but not for d = N. Such classes include domains with polynomial outward pointing cusps, where d depends on the type and the sharpness of the cusp (see, for instance, [1, thm 5.35]). Other such classes are described in [16, § 1.4.5].

Note that domains satisfying an interior cone condition are not necessarily extension domains. As an example, consider a domain with an inward pointing cusp, or a slit domain. On the other hand, extension domains do not need to satisfy an interior cone condition as the example of a fractal domain shows.

3.3. The third boundary-value problem

In the third boundary-value problem, we have T3 = aD. As for the second boundary-value problem, we choose V = Wi(D). If we assume that D is Lipschitz, then proposition 3.1 shows that V satisfies assumption 2.1. We further set 8B = 1, so that

Ilullwj(n) ~ qB(U)

for all u E V. We saw in the previous discussion on the second boundary-value problem that every Lipschitz domain is an extension domain and therefore admits the Sobolev inequality. Hence theorem 2.5 applies with d = N if N ~ 3, and d E (2,00) if N = 2. The constant CB depends on the shape of the domain, as discussed before.

If we assume that ess-inf bo ~ (30 for some (30 > 0, there is an alternative approach leading to a very weak dependence on the domain. Under this assumption, we define an equivalent norm

(3.1)

on Wi(D). An inequality due to Maz'ja (see [16, corollary 4.11.1/2]) asserts that

lI u lbN/(N-l) ~ c(N, IDI)llullv (3.2)


for all u E Wi(D), where the constant c(N, IDI) only depends on the dimension N and an upper bound for the measure of D. We now choose 58 := 0 and set d:= 2N. Then, due to the above inequality, we easily see that

Ilull~d/Cd-2) ~ ao max{ ao!' /301 }c(N, IDI)2q~( u)

for all u E wi (D). This implies that

ao1c~ = max{ ao1, /301 }c(N, IDI)2.

Hence the constants in the estimates given in theorem 2.5 only depend on N, p and the upper bounds for max{ ao1 ,/301} and the measure of D. As the estimates do not depend on the shape of the domain, one should be able to find a weak formulation for the third boundary-value problem that makes sense for general domains. This can, in fact, be done. The idea is to introduce a different Hilbert space V. The inequality (3.2) holds for all u E Wi(D) n COO(D) n C(tl) for which the right-hand side is finite. We define V to be the completion of this set with respect to the norm (3.1). In [16, §§ 3.6 and 4.11], this space is extensively discussed and denoted by Wi2(D,aD). Examples show that for irregular domains this space may be smaller than Wi(D) (see [9, remark 3.5(f)]). For more about the space wi 2(D, aD) and a comprehensive discussion of the third boundary-value problem on i~regular domains, we refer the reader to [9]. Theorem 2.5 also applies to that case, and we recover some of the results in [9]. Applications of these domainindependent estimates to problems with varying domains, including some boundary homogenization problems, can be found in [7].

3.4. Mixed boundary conditions

Let D be a bounded open set and suppose that Ti (i = 1,2,3) are open and closed, with r3 Lipschitz. Define V as in proposition 3.1 and therefore V satisfies assumption 2.1. We now want to see how to get Sobolev inequalities for V. Select a neighbourhood U of r2 U r3 with a smooth boundary not intersecting r1. Then assume that Wi(D n U) admits a Sobolev-type inequality (2.13) for some d > 2. Conditions for this to be true are discussed in § 3.2. Note that d only depends on the smoothness properties of r2 . Next let cp E Crgo(U) be such that u = 1 in a neighbourhood of r2 U r3 and 0 ~ cp ~ 1. Clearly, u = cpu + (1 - cp)u. By our assumption on D n U and the Sobolev inequality for functions in Wi(D), we see that

Ilcpu I12d/Cd-2) ~ c1I1V(cpu)112 ~ c1(1 + IIVCPlloo)lluIIWiCst),

11(1- cp)uI12d/Cd-2) ~ c21IV((1- cp)u)112 ~ c2(1 + IIVcplloo)lIuIIWiCst)·

Choosing 58 = 1, we therefore get

for all u E V, that is, V admits a Sobolev-'-type inequality. The constant C1 depends on r2 , r2 and the neighbourhood U, whereas C2 depends only on N, and if d > N also on the measure of D. Alternatively, one could work with 58 = 0 if either r 1 f. 0 or r3 f. 0 and bo f. 0 on some set of positive surface measure.

802 D. Daners

4. Abstract regularity results

We assume throughout that V and H are Hilbert spaces over IR (or C) such that V is densely and continuously injected into H. By means of the Riesz isomorphism, we can identify H with its anti-dual space H'. By the assumptions on V and H, we therefore get, by duality, that

Vc........tHc........tV' ,

where V'is the anti-dual space of V. Suppose now that a : V x V -+ IR (or C) is a sesquilinear form. Recall that a(·, .) is said to be bounded if there exists a constant M ~ 1 such that

la(u, v)1 ~ Mllullvllvllv (4.1)

for all u, v E V. It is said to be V -elliptic if there exist constants Q > 0 and AD > 0 such that

Qllulltr ~ Rea(u, u) + Aollull~ (4.2)

for all u E V. Finally, note that if a(·, .) is a bounded sequilinear form on V, then there exists an operator A E £(V, V') such that

a(u, v) = (Au, v) (4.3)

for all u, v E V. We call A the operator induced by a(·, .). Indeed, if we fix u E V, then the functional mapping v E V to a( u, v) is an element of V'. As every Hilbert space is reflexive, there exists a unique W E V' such that a(u, v) = (w, v) for all v E V. We then set Au := w. By definition of the dual norm and (4.1), we have

IIAullvl = sup I(Au,v)1 = sup la(u,v)1 ~ Mllullv, Ilvllv=1 IIvllv=1

which shows that A E £(V, V'). We will study properties of solutions to the abstract equation

Au=j, (4.4)

with j E V'. Clearly, equation (4.4) holds if and only if a( u, v) = (j, v) for all v E V. Applied to the elliptic boundary-value problems discussed in § 2, this shows that weak solutions are the solution of an abstract equation and vice versa.

We now establish a priori estimates for solutions to the abstract equation (4.4). We consider the particular case where H = L 2 (X), with (X,I-£) being an arbitrary measure space. Throughout we assume that V c........t L 2 (X) and that V is dense in L2(X), We further assume that A E £(V, V') is the operator induc~d by the bounded sesquilinear form a(·,·) on V. Finally, for every measurable function u : X -+ IR (or C) and every constant q ~ 2, we define

uq := sgn ulul q-

1, (4.5)

where, as usual, sgn~ := ~I~I-l for ~ E C \ {O}, and sgnO := O.

ASSUMPTION 4.1. Suppose that Cll C2 : [2,00) -+ [1,00) are non-decreasing functions of at most polynomial growth, with Cl (2) = C2 (2) = 1. We further assume that there exist constants a > 0, Al ~ 0 and d > 2 such that

allull~d/(d_2) ~ Cl (q) Re(Au + C2(q)Al U, uq )

for all V and all q ~ 2 whenever the right-hand side is finite.

(4.6)


REMARK 4.2. In case of elliptic boundary-value problems, the above assumption derives from the structure of the elliptic operator and the Sobolev-type inequality (2.13). For details, see § 5.

REMARK 4.3. Note that in this section it is not necessary that the operator A is induced by a sesquilinear form a on V. It is sufficient to assume that A is a closed operator on L2 (X) .

PROPOSITION 4.4. Suppose that u E V satisfies Au = f, with f E Lp(X) n L2(X) for some p E [2, (0). Further suppose that assumption 4.1 holds. Then u E Lp(X) and

Ilull~d/(d-2) :::;; a-lcl (p )C2(P )(lIfllp + AIilull p) Ilull~-l. (4.7)

Proof. First note that, as f E Lp(X) n L2(X), we have, by interpolation, that f E Lq(X) for all q E [2,p]. Now fix an arbitrary q E [2,p] and assume that u E Lq(X). Then it follows from assumption 4.1 that

allull~d/(d_2) :::;; Cl(q) Re(Au + C2(q)AlU, uq)

= Cl (q)(Re(f, uq) + C2(q)Alllull~) :::;; Cl (q)(llfllqllull~~~_l) + C2(q)AIilull~) :::;; Cl(p)C2(q)(lIfll q + AIilullq)llull~-l. (4.8)

Therefore, u E Lq(X) with q E [2,p] implies that u E Lqd/(d-2) (X). Recall that d> 2, and thus the sequence qn defined by

qo:= 2 and qn+1:= (d: 2J

for all n E N goes to infinity as n tends to infinity. By definition (4.5), we know that uqo ,= U2 = u, and thus uqo E L2(X). An iteration of (4.8) starting with qo = 2 therefore shows that u E Lqn (X), as long as qn-l E [2,p]. As qn grows to infinity, there exists n E N such that qn E [2,p] and qn+l > p. As we already know that u E L2(X), we have, by interpolation, that U E Lp(X). To get (4.7), we simply choose q = p in (4.8), concluding the proof of the proposition. D

THEOREM 4.5. Suppose that uE V, with Au = f, and that f E Lp(X) n L2(X) for some p ~ 2. Further suppose that (4.6) is satisfied for all q ~ 2. Then there exists a constant C ~ 1, depending only on p, d, Cl and C2, such that

(4.9)

if p E (2, ~d) and

. '{Ca-l(llfllp + AIilullp) + lIullp if J-L(X) is arbitrary, Ilull oo

:::;; Ca- l J-L(X)(2P-d)/dp(llfll p + AIilull p) if J-L(X) < 00 (4.10)

ifp > ~d.

804 D. Daners

Proof. To prove (4.9) and (4.10), we iterate an inequality derived from (4.6). We know already from proposition 4.4 that, under our assumptions, u E Lp(X). It follows from (4.6), the assumption that f E Lp(X) and Holder's inequality that

provided q ~ p. Here, p' := p/(p-1) is the dual exponent to p. Note that p'(q-1) ~ 2 for that range of q. If we set c(q) := Cl(q)C2(q) and

v := a(llfllp + Alllullp)-lU,

the above inequality implies that

whenever the right-hand side is finite. The idea now is to iterate the above inequality, starting with qo := p. We define qn inductively by qnd/(d - 2) = p'(qn+l - 1) for all n E N. If we solve this identity for qn+b we get

where d(p- 1)

'f/:= (d - 2)p·

Observing that qn - 1 = 'f/qn-l, the above inequality reads

(4.11)

( 4.12)

The above inequality tells us that u E Lpl(qn-l) (X) implies that u E Lpl(qn+l-1) (X). We claim that qn is strictly increasing, so that (4.12) can be iterated. To prove this, we use the definition of 'f/ to observe that p = qo < 1 + 'f/p = ql. Assuming that qn-l < qn, it is quite obvious that qn+l = 1 + 'f/qn > 1 + 'f/qn-l = qn. Hence our claim follows by induction. We now prove that

n

Ilvllqnd/(d-2) = Ilvllpl(qn+l-1) ~ II C(qk)ryn-k /qn Ilvlli;-l)ryn /qn (4.13) k=O

for all n E N. Forn = 0, the above inequality follows immediately from (4.12) by substituting qo = p. Assume that the above inequality holds for some n ~ O. Then we get, from (4.12) and the induction assumption, that

(

n )ryqn/qn+l Ilvl!qn+l d/(d-2) ~ C(qn+l)l/qn+l !! C(qk)ryn-k /qn Ilvlli;-l)ryn /qn

n+l = II C(qk)ryn+l-k /qn+lll v lli;-l)ryn+l /qn+l,

k=O

which is exactly what we want. We now distinguish two cases, namely p < ~d and p > ~d. They lead to (4.9) and (4.10), respectively.


We first assume that p E [2, ~d) and prove (4.9). We start by computing the limit of the sequence (qn). To do so, we need the explicit representation

n-l

qn =prt + L'TIk (4.14) k=O

of qn, which follows from the recursion formula (4.11) by induction. Using the definition of'TI, we see that 0 < 'TI < 1 if and only if p < ~d. Hence, passing to the limit in (4.14), we obtain

_. 1 p(d-2) q:= hm qn = -- = .

n-+oo 1 - 'TI d - 2p (4.15)

By assumption, the function c : [2,00) ---+ [1,00) is non-decreasing in q, and thus we conclude from (4.13) that

Il v ll qn d/(d-2) :::;; C(qn)2:~=o ryk /qn Ilvlli;-l)ryn /qn.

We now pass to the limit. As 0 < 'TI < 1,we get from (4.15) that

n k

lim L!L = 1 and n-+oo k=O qn

'TIn lim - =0.

n-+oo qn

This implies that

lim sup IIvllqnd/(d-2) :::;; Ilvllqd/(d-2) = Ilvlldp/(d-2p) :::;; c(q). n-+oo

Taking into account the definition of v, the estimate (4.9) follows immediately. We now assume that p > ~d and prove (4.10). In this case, note that 'TI > 1. We

first determine the behaviour of (qn). We claim that

(4.16)

for all n E N. We give a proof by induction. As qo = p, this is obvious for n = O. Assume now that (4.16) holds for some n ~ O. As 'TI > 1, we then conclude from the recursion formula (4.11) for qn that

'TIn+lp:::;; 'TIqn < 1 + 'TIqn = qn+l < 2'T1qn :::;; (2'T1t+ 1p.

Hence (4.16) follows by induction. We assumed that Cl and C2 grow at most polynomially. Hence the same is true for C = Cl C2. We therefore find constants {3 ~ 1 and 'Y ~ 0 such that c(q) :::;; {3q'Y for all q ~ 2. Substituting this into (4.13), we see that

n

II v ll qnd/(d-2) :::;; II ({3qnryn-k /qn Ilvlli;.-l)ryn /qn k=O

for all n E N. Using (4.16), we therefore conclude that

Il v ll qn d/(d-2) :::;; (2{3'T1p)'Y2:~=o kryn-k /qn Ilvlli;-l)ryn /qn (4.17)

806 D. Daners

for all n E N. Next we consider the exponents on the right-hand side of the above inequality. Using (4.16) and the fact that TJ > 1, we see that

n k n-k n

I' L _TJ_ :::;; 1 lim L kTJ- k < 00

k=O qn p n-+oo k=O

for all n E N. Using the representation (4.14) for qn and the fact that TJ > 1, we conclude that

lim '!L = lim p+ ""'TJ- k = p+ -- = 1- - (p-l)-l, n (n )-1 ( 1 )-1 ( d ) n-+oo qn n-+oo ~ TJ - 1 2p

k=1

where we used the definition of TJ to get the last equality. Passing to the limit in (4.17), we see that

Ilvll oo = 2~~ Il v ll qn d/(d-2) :::;; (2,6TJP)')'8/Pllvll~-d/2p.

Applying Young's inequality, we see that Ilvll oo :::;; C + Ilvllp for some constant C depending only on d, p, Cl and C2. Using the definition of v, the claim for arbitrary X follows. If X is of finite measure, then, by Holder's inequality, Ilvllp :::;; j-t(X)I/P llvll oo

and thus Ilvll oo :::;; Cj-t(X)(2p-d)/dp for some constant C only depending on d, p, Cl

and C2. This concludes the proof of the theorem. 0

For some purposes, it is useful to replace Ilulip by IIul12 in the above estimates.

COROLLARY 4.6. Suppose that the assumptions of the above theorem are satisfied. Then there exists a constant C ~ 1, depending only on d, p, Cl and C2, such that

Il u lldp/(d-2p) :::;; C( a-1 1lfllp + (a7"1 Al)l+d(p-2)/4p Ilu112) (4.18)

if p E (2, ~d). For p > ~d, we have

( 4.19)

if X is arbitrary and

Ilull oo :::;; C( a-I j-t(x)(2p-d)/dP llfll p + a-p/2 j-t(X) (2p-d)/2d Atliu112)

if j-t(X) < 00.

Proof. By a well-known interpolation inequality (see [11, p. 147]), we have, for 2:::;; p:::;; q:::;; 00,

( 4.20)

where j-t:= (~ - l/q)(I/p - l/q)-I. Assume now that p E [2, ~d) and that q = dp/(d - 2p). Then j-t = 1 + d(p - 2)/4p and if we choose c := a/2AIC, we conclude from (4.9) and (4.20) that

~ Il u lldp/(d-2p) :::;; Ca-1 1lfllp + (2Ca- 1 Al)d(p-2)/4P ll u I1 2.

By changing the constant C, inequality (4.18) follows. If p > ~d, we choose q := 00

and observe that j-t = ~p. We can then proceed similarly as before. We leave the details to the interested reader. 0


COROLLARY 4.7. Suppose that f E L2(X) n Lp(X) for some p ~ 2. Then, for all A > AI, the problem

AU+AU = f (4.21)

has a unique solution u E V . Moreover, there exists a constant c such that

Ilullm(p) ~ c(llfllp + (A - Al)-11IfI12),

where m(p) := dp/(d - 2p) if p E [2, ~d) and m(p) = 00 if p > ~d. Finally, the constant c only depends on d, p, Cl, C2 and upper bounds for a-I and AI.

Proof. We first show that (4.21) has a unique solution u E V if A > AI. To do so, we first show that for every A > Al there exists al > 0 such that aIilull~ ~ a(u, u) + Allull~ for all u E V. Suppose now that A > AI. As a(·,·) is assumed to be V-elliptic, there exists Q > 0 and AO ~ 0 such that (4.2) holds for all u E V. If AO ~ AI, our claim follows. Now suppose that Al < A. Setting q = 2 in (4.6), we get (A-Al)llull~ ~ a(u,u)+Allull~ for all u E V and A-AI> O. Using the V-ellipticity, we thus obtain

Qllull~ ~ a(u,u) + Aollull~ = a( u, u) + Allull~ + (A - AO) Ilull~ ~ a(u, u) + Allull~ + (A - AO)(A - Al)-l(a(u, u) + Allull~) = (1 + (A - AO)(A - Al)-l )(a(u, u) + Allull~)

for all u E V. This shows that a(·,·) is V-elliptic for all A > Al for some al > O. Therefore, the Lax-Milgram theorem (see [22, §III.7]) implies that (4.21) has a unique solution for all A > AI. If u is the solution of (4.21), it also follows that

o ~ a(u,u) + AIilull~ = (f,u) - (A - Al)lIull~ ~ IIfl1211ul12 - (A - Al)llull~·

Thus II u 112 ~ (A - AI) -111 f 112. To get the last assertion of the corollary, we only need to substitute this estimate into (4.18) and (4.19), respectively. D

REMARK 4.8. The above corollary shows that A + A has a bounded inverse from Lp(X) n L2(X) to Lm(p)(X) for all A > AI. If J.L(X) < 00, then this shows that A E Q( -A) and (A + A)-l E £(Lp(X), Lm(p) (X)) for all A> AI.

5. Proof of the estimates for elliptic equations

The purpose of this section is to show that the elliptic boundary-:-value problems considered in § 2 fit into the theory developed in § 4. As a consequence, theorem 2.5 and corollary 2.6 follow. The main point is to establish some basic estimates for the form a(·,·) associated with (A, B) (for the definition, see (2.5)). In particular, we verify assumption 4.1, provided V admits a Sobolev-type inequality. This shows that theorem 2.5 applies. In the sequel we assume that V is a Hilbert space satisfying assumption 2.1. We first observe that a(·,·) is bounded on V.

LEMMA 5.1. Under assumption 2.1, the form a(·,·) associated with (A, B) is bounded on V.

808 D. Daners

Proof. Using the norm (2.8) on V, it is easily seen that a(·, .) is bounded. D

Next we prove a general inequality, which will lead to the key inequality (4.6) in the abstract theory, and to the V-ellipticity of the form a(·, .). Before we state the inequality, we define, for every k, q ~ 1, the function gk,q : ffi. ---7 ffi. by

(c) := {sgn ~1~lq if I~I :::; k, gk,q ~ kq-l~ if I~I ~ k (5.1)

for all ~ E ffi..

LEMMA 5.2. Let u E V and set Vk,q := gk,q-l 0 u and Wk,q := gk,q/2 0 u. Further, let 60 be given by (2.10). Then Vk,q, Wk,q E V and

~(aoll\7wk,qll~ + Ilwk,q00lli2(r3)) :::; ~q(a(u,Vk,q) + (q -1)60(u,Vk,q)) (5.2)

for all k ~ 1 and q ~ 2.

Proof. It is obvious that gk,q is smooth except at ~ = ±k, that gk,q(O) = 0 and that g~,q E Loo(ffi.). By the last part of assumption 2.1, we have gk,q 0 u E V for all u E V. It follows from (2.6) and [11, thm 7.8] that \7(gk,q ou) = (g~,q ou)\7u, where we set g~,q(~) = 0 for ~ = ±k. To simplify the notation, let us fix q ~ 2 and k ~ 1 arbitrary, and set v := Vk,q and W := Wk,q. According to the previous reasoning, v, W E V. An elementary calculation shows that

q2 OjWOiW = 4(q _ 1) OjUOiV (5.3)

and

(5.4)

at all points where lui < k. At all points for which lui> k, we obtain

(5.5)

and (5.6)

Now, if lui < k, the ellipticity condition (2.2), (5.3) and (5.4) imply that

N

aol\7wI2:::; L aijOjWOiW i,j=l

2 N

4( q- 1) L aijOjUOiV

q i=l .

,;; 4(qq~ 1) (t,(taiAu + aiU)8iV + (t,b i8iU + COU) V)

+ ~q(lla + bll oo )l\7wllwl + ~qllcolloow2.


Recall the elementary inequality yz :::;; ~6"y2 + y2/26", which holds for all y, z ~ 0 and 6" > O. Applying this inequality to the term involving Iwll\7wl and bringing ao2-11\7wI 2 to the left-hand side, the above inequality yields

!ao!\7w!20( 4(qq~ 1) (t, (t,aiAU + aiU)8iV + (t,b i 8iU + CoU)V)

+ ~q2I1a; bll~ Iwl 2 + ~qllcolloolwl~. ao

Clearly, the expression on the right-hand side of the above inequality is nonnegative. As q(q - 1)-1 :::;; 2 for q ~ 2, we therefore get

N N N

!ao!\7w!2 0( !q(t;(~aij8jU + aiU )8iV + (t;bi8iU + COU) V + (q -1)8ouv), (5.7)

where we also used that w 2 = uv. For lui > k, we proceed similarly. First we observe that q2 / 4( q - 1) ~ 1 for all q ~ 2. Thus, using the ellipticity condition (2.2) and (5.5), we get

for all q ~ 2. Then we apply a similar procedure as for lui < k, using (5.6) to get (5.7). As \7v = \7w = 0 at all points where lui = k, inequality (5.7) holds trivially. Hence (5.7) holds in all of n. If we integrate over n and add 2-1I1wJbOlli2cr3) on both sides of the inequality, the assertion of the lemma follows immediately. 0

The above lemma, in particular, implies that a(·, .) is V -elliptic, and thus can be considered as some sort of higher V-ellipticity.

COROLLARY 5.3. The form a(·,·) associated with (A,B) is V-elliptic. More pre-cis ely,

aollull~:::;; a(u,u) + (bo + ao)lIull~ (5.8)

for all u E V, where 11·11 B is the equivalent norm on V defined by (2.8).

Proof. Observe that gk,l ou = u for all k E JR. Choosing q = 2 and adding 2-1ao lIull~ to both sides of (5.2), inequality (5.8) immediately follows. 0

Another immediate consequence of lemma 5.2 is the following inequality involving the seminorm qB defined by (2.11).

COROLLARY 5.4. Let qB and)..-8 be defined by (2.11) and (2.12), respectively. Then

~aOqB(wk,q) :::;; ~q(a(u,Vk,q) + (q -l)AB(u,Vk,q))

for all u E V and all k ~ 1 and q ~ 2.

Due to the abstract theory at the start of § 4, we know that there exists an operator A E £(V, V') satisfying a(u, v) = (Au, v) for all u, v E V. Also recall that

810 D. Daners

we have set uq := sgn ulul q- I for q ;::: 2. In the following proposition, we show that assumption 4.1 is satisfied for elliptic boundary-value problem.

PROPOSITION 5.5. Let u E V such that Au E Ls(fl) for some s ;::: 2. Further, let u E Lq(fl) n LS/(q_1)(fl) for some q ;::: 2. Finally, assume that there exist constants d> 2 and CB > 0 such that the Sobolev-type inequality (2.13) holds. Then

2:11Iu"~d/(d-2) ~ ~q(Au + (q - I)ABu, uq). (5.9)

Proof. By corollary 5.4, the assumption that a Sobolev-type inequality holds and the definition of A, we have

2a~ Ilwk,qll~d/(d-2) ~ ~aOq§(wk,q) ~ ~q( (Au, Vk,q) + (q - I)AB(u, Vk,q)) (5.10) cB

for all k ;::: 1. Note that Igk,q 0 ul is non-decreasing in k and that it converges to lul q

as k goes to infinity. By the monotone convergence theorem, we thus have

kl~~ Ilwk,qll~d/(d-2) = Illulq/2111~d/(d_2) = Ilull~d/(d-2)"

As u E Lq(fl) by assumption, (u, Vk,q) converges to (u, uq) as k goes to infinity. Next note that u E Ls/(q-l) (fl) implies that uq E LSI (fl) and therefore (Au, Vk,q) converges to (Au, uq) as k goes to infinity. Letting k go to infinity in (5.10), we thus arrive at (5.9), concluding the proof of the proposition. D

Theorem 2.5 and corollary 2.6 now follow from theorem 4.5 and corollary 4.7, respectively. We only need to observe that Al := AB, a := ao/2c§, CI(q) := ~q and C2(q) := q - 1.

6. A converse to the regularity results

In § 4 it is shown that a Sobolev-type embedding theorem implies Lp-Lq-estimates for solutions to abstract elliptic equations. In this section we want to show that, at least for self-adjoint problems and finite measure spaces, such estimates imply a Sobolev-type inequality. In fact, we get exactly the exponent d we would expect from p and q. We then apply the result to prove theorem 2.8. To prove our result, we use the same setup as in § 4, but assume that J-l(X) < 00 and that a(·,·) is hermitian, that is, a(u,v) = a(v,u) for all u,v E V. As a consequence, the operator A induced by a(·,·) is self-adjoint on L2(X).

THEOREM 6.1. Suppose that A is :self-adjoint, that J-l(X) < 00 and that, for some X E ~ and 1 < p < q < 00,

(A + A)-I E £(Lp(X), Lq(X)).

Then V y L2d(d-2)-1 (X) for d := 2pq(q - p)-I.

The proof will be given in two lemmas. The first depends on the problem being self-adjoint. It remains true if A and its adjoint have the same smoothing property. As the second lemma does not depend on the operator being self-adjoint, this means that the assertions of the above theorem remain true for that case.


LEMMA 6.2. Suppose that the assumptions of theorem 6.1 are satisfied. Then

(A + A)-l E £(LSI(X), Ls(X)),

where s := 2pq(pq + p - q)-l > 2, and s' := s/(s - 1) is the dual exponent to s.

P~~of. As the problem is self-adjoint, we get, by duality, that

(A + A)-l E £(Lp(X), Lq(X)) n £(Lql(X), Lpl(X)).

lret r, s E (1,00) be defined by l/r = 1/2p + 1/2q' and 1/ s = 1/2q + 1/2p'. Then

'1- ~ = 1- ~~ - ~ (1- ~) = ~ + ~ (1- ~) - ~ (1- ~) = ~~ +~! = ~. s 2 q 2 p 2 2 q 2 p 2 q' 2 p' r

and thus r = s'. Applying the Riesz-Thorin interpolation theorem [3, thm 1.1.1]), we therefore conclude that

(A + A)-l E £(LSI (X), Ls(X)).

By the definition of s, we have s = 2pq / (pq + p - q). By assumption, p < q and thus pq > pq+p-q. As p, q > 1, we see that pq+p-q > 0 and thus 1 < pq(pq+p_q)-l. Hence s > 2, and the proof of the lemma is complete. 0

The next lemma proves an embedding for the Hilbert space V. Its proof does not depend on whether the problem under consideration is self-adjoint or not.

LEMMA 6.3. Suppose that, for some A E IR and s > 2,

(A + A)-l E £(LSI (X), Ls(X)).

Then V Y Ls(X).

Proof. First note that, without loss of generality, we can assume that A = O. Then, by definition of A and our hypotheses, we have

A-I E £(V', V) n £(LSI(X),Ls(X))

Hence

aliA -lull~ ::( a( u, u) = (u, A -lu)::( Ilulls'IiA -lulls ::( IIA -lll.c(Lsl ,Ls) Ilull;, fqr all u E L2 (X) nLsl(X). As L2 (X) nLsl(X) is dense in LSI(X), we conclude that A-I E £(LSI(X), V). We already know that A E £(V, V'), and thus i := AA-l E

£(LSI(X), V'). Note that i is the embedding of LSI (X) into V', so we have proved that LSI(X) Y V'. Observe that LSI (X) n L2 (X) is dense in V', and so by duality V Y Ls(X), as claimed in the lemma. 0

Proof of theorem 6.1. The assertions of theorem 6.1 are an immediately consequence of lemmas 6.2 and 6.3. We only need to compute s in terms of d. We set s = 2d(d - 2)-1 and solve for d. Then d = 2s(s - 2)-1 and an elementary calculation using the value of s given in lemma 6.2 shows that

d = 2 . 2pq 2pq 2pq 2pq - 2 (pq + p - q) pq - pq - p + q q - P

This concludes the proof of the theorem. o

812 D. Daners

We showed in § 5 that the elliptic boundary-value problems under consideration fit into the framework of § 4. Hence theorem 2.8 follows from theorem 6.1.

7. Compactness

In this section we prove the compactness of the resolvent as an operator on Lp(fl) for all p E [2,00), provided the assumptions of theorem 2.5 hold. Note first that the Sobolev-type inequality (2.13) implies that V is continuously embedded into L2d/(d-2) (fl). The key to the compactness is the following observation concerning the compactness of the embedding of V into L2(fl).

LEMMA 7.1. Suppose that fl c JRN is an open set of bounded measure and that p E (1, N). Further suppose that V 4 wi (fl) is a subspace (with a possibly stronger norm) and that V 4 Lq(fl) for some q E (2, 2N/(N - 2)). Then the embedding V 4 Lr(fl) is compact for all r E [1, q).

Proof. Suppose that B is a bounded subset of V. By assumption, we have that V 4 Wi(fl) n Lq(fl), which implies that B is a bounded set in Wi(fl) n Lq(fl). Hence it suffices to prove that every bounded set in wi (fl) n Lq (fl) is relatively compact in Lr(fl) for all r E [0, q). To show this is the case, it is sufficient to show that every sequence (Un)nEN converging to zero weakly in wi(fl)nLq(fl) converges to zero in Lq(fl) strongly for all r E [1, q). Fix c > 0 and r E [1, q) and choose a subset fl' c fl whose closure is contained in fl, such that I fl \ fl'l < c. It is well known that the embedding wi (fl') 4 Lq(fl') is compact for all q E (2, 2N/(N -2)) (see [1, thm 6.2]). Hence Un converges to zero in Lq(fl') strongly. Hence, by Holder's inequality,

We showed already that the first term converges to zero as n goes to infinity. As (Un)nEN is weakly convergent in Lq(fl), the sequence is bounded. Hence the second term can be estimated by Ifl \ fl'll/r-l/qllunIILq(n) ::( cc1/ r - 1/ q for some constant c > o. As c > 0 was arbitrary, this shows that Ilunllr,n can be made arbitrarily small, showing that Un converges to zero in Lr(fl). 0

We are now in a position to prove theorem 2.9. Suppose that).. > )..B and that A E £(V, V') is the operator induced by the form a(·,·) associated with (A, B). According to remark 2.7, we have that A E £(Lp(fl)) for all p E [2,00] (see § 4 for the definition of A). We have the following commutative diagram

V ~ L2(fl)

1 (>'+A)-l 1 (>'+A)-l

v' ~ L2(fl)

where i is the natural inclusion. As the natural inclusion is compact by lemma 7.1, we conclude that ()"+A)-l E £(L2(fl)) is compact. We also know that ().. + A)-l E £(Loo(fl)). Hence, due to a compactness property of the Riesz-Thorin interpolation (see [15]), it also follows that ().. + A)-l E £(Lp(fl)) is compact for all p E [2,00).


By the resolvent identity, it follows that the resolvent is compact for all A E p( -A). This completes the proof of theorem 2.9.

Acknowledgments

This work was carried out while the author visited the Institute of Mathematics at the University of Zurich and the Institute for Mathematical Research (FIM) at the ETH Zurich, Switzerland. Their generous support is gratefully acknowledged.

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a priori estimates for solutions to elliptic equations on non-smooth domains

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