Existence of solutions for semilinear elliptic boundary value problems on arbitrary open sets Текст научной статьи по специальности «Математика»

YflK 517.956.2

BecTHHK Cnöry. Cep. 1. T. 2(60). 2015. Bun. 4

EXISTENCE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS ON ARBITRARY OPEN SETS

Reinhard Stahn

Institut für Analysis, Technische Universität Dresden Germany, 01062, Dresden, Helmholtzstr., 10

We show the existence of a weak solution of a semilinear elliptic Dirichlet problem on an arbitrary open set Q. We make no assumptions about the open set Q and very mild regularity assumptions on the semilinearity f, plus a coerciveness assumption which depends on the optimal Poincare—Steklov constant Ax. The proof is based on Schaefer's fixed point theorem applied to a sequence of truncated problems. We state a simple uniqueness result. We also generalize the results to Robin boundary conditions. Refs 17.

Keywords: elliptic, semilinear, locally convex, fixed point, arbitrary domain.

1. Introduction. The question of existence of (weak) solutions for a boundary value problem like

—Au(x) = f (x, u(x), Vu(i)) (x G 9), (__)

Bu(x) =0 (x G d9) (1)

is a classical problem. Here Bu(x) = 0 (x G d9) is simply an abbreviation for specific boundary conditions (eg. Dirichlet, Neumann or Robin boundary conditions). However, in older publications (e.g. [1-3] and references therein) it seems to be standard to assume that 9 is bounded and has sufficiently smooth boundary. On the other hand for a long time it is well known that Dirichlet (and Neumann) boundary conditions can easily be defined on arbitrary open sets in a weak sense. But until now a general existence theorem for weak solutions of (1) on arbitrary open sets (that means in particular possibly unbounded, not smooth, not connected), possibly irregular semilinearities f (that means in particular possibly not Lipschitz continuous in u or Vu) and general boundary conditions seems to be unknown.

In this paper we state and prove such a theorem with the help of a fixed point theorem in a locally convex space. For simplicity we restrict ourselves to the case of Dirichlet boundary condition. A short discussion on how the results extend to other boundary conditions is included at the end of the paper. To find solutions of (1) with the help of fixed point methods is standard. However, in the majority of textbooks a Banach space setting is presented (cf. [4-6]).

2. Assumptions and main results. Let 9 Ç be an arbitrary open set. We consider the following Dirichlet problem:

—Au(x) = f (x, u(x), Vu(x)) (x G 9), (2)

u(x) =0 (x G do). (2)

We call u a weak solution of this problem if u G H0(9) such that f (x, u, Vu) G L2(9) and

f Vu ■ V^dx = f f (x, u, Vu)^dx V^ G C°°(9).

JQ JQ

In the following we show under which assumptions on the semilinearity f we can prove a priori bounds, existence and uniqueness of weak solutions.

2.1. Assumptions and notation. We assume that f : Q x R x Rd ^ R is a Caratheodory function. This means that f = f (x, s,£) is measurable as a function in x when s and £ are fixed, and is jointly continuous as a function in s and £ when x is fixed, for almost all x. The function f should satisfy a coerciveness and a growth condition

f (x, s, £)s < (Ai - e)s2 + L |£s| + h(x) |s| Vx G Q, s G R, £ G Rd, (3)

f (x, s, £)s >-7(|s|) |s| — Lo |£s| — ho(x) |s| Vx G Q, s G R, £ G Rd. (4)

Here A1 is the optimal Poincare—Steklov constant for the Dirichlet—Laplace operator, i.e. A1 > 0 is the largest real number such that A1 ||u||L2(Q) < ||Vu||L2(Q) is true for all u G Hq(Q). The positive constant L has to satisfy

L < Lmax(e, Ai) :-

e/VXT if e < 2Ai, 2Ve - Ai ife>2Ai.

(5)

Furthermore e > 0 and h > 0 with h G L2 n Lq(Q) for some q > 2 which (for simplicity) is not equal to d/2. The number q will serve as a parameter. L0 > L is an arbitrary constant and h0 G L2(Q) with h0 > 0. The monotone increasing function 7 : [0, to) ^ [0, to) is assumed to satisfy

and in case of q < d/2 also

Y (s)

iim sup- < 00

s^Q S

lim sup

7(g)

sq**/2

<

(6)

(7)

for the positive real number q** = qd/(d — 2q). For future use we also define q** = to if q > d/2. The bigger the parameter q, the more restrictive is the coerciveness condition (3) but the less restrictive is the growth condition (4). Note that the condition (6) is not needed if the measure of Q is finite. In the case when (6) is not satisfied but the measure of Q is finite we could change y(s) to (y(s) — 7(1))+ and add the additional constant 7(1) to the function h0 without touching its L2-integrability. Here and in the following for a real number a we define a+ := maxja, 0}.

We will also consider the following two Dirichlet problems on Q:

—Av = (A1 - e)v - L |Vv| — h(x) (x G 0),

v(x)

0

(x G dO)

(8)

and

—Av = (A1 — e)v + L |Vv| + h(x) (x G O),

v(x)

0

(x G dO).

(9)

In Section 4.4 we prove that solutions of these two equations exist and are unique. Therefore by y_,v let us denote the solutions of (8) and (9), respectively. We will also see that v < 0 < v. By (3) this implies —Av < f(x,v,Vv) and —Av > f(x,v,Vv) that is, these two functions are sub- and supersolutions of (2).

s

00

2.2. Main results. The first step to prove existence of a weak solution of (2) is to prove a priori estimates for hypothetical solutions of a class of semilinear problems: Let y > max{0,e — Ai}, 0 < t < 1 and w be an open subset of 9. Moreover, let v0 < 0 < vg be two measurable functions on 9. Consider

{—Am + yu = t (f (x, u), Vu) + ya(x, u)) (x) (x G 9), ^ v ^

(10)

u(x) =0 (x G d9),

where <r(x, s) = max{v0(x), min{s, v1(x)}}.

Theorem 1 (a priori bounds in Hg). Assume that f satisfies (only) the coerciveness condition (3). Let u be a weak solution of (10). Then u satisfies an a priori estimate in Hg(9). More precisely, there exists a constant C > 0 such that

IMIh(n) < C IHIl^) •

The constant depends only on Ag,e and (Lmax(e, Ag) — L)-1.

This theorem is the central argument in the proof of the existence of a weak solution of (2) in Section 5.

Theorem 2 (a priori bounds in Lq»»). Assume that f satisfies (only) the condition (3), where the term Ag —e is replaced by any positive number 6. Let u be a weak solution of (10). Then u satisfies an a priori estimate in Lq»»(9). More precisely, there exists a constant C > 0 such that

The constant depends only on d, q, 6 and L.

Much more interesting than merely a bound for the Lq»»-norm is the fact that weak solutions are even a priori dominated by functions in the Lq»» (9).

Theorem 3 (a priori domination in Lq»»). The Dirichlet problems (8) and (9) have unique weak solutions v < 0 andv > 0 respectively. Moreover, if we assume that f satisfies (only) the coerciveness condition (3) then

v < u < v

for every weak solution u of (10).

The main result of this paper is

Theorem 4 (existence). Under the assumptions in Section 2.1 the Dirichlet problem (2) admits a weak solution.

Note that only in this theorem, but not in the preceding three theorems we assume the validity of the growth condition (4). The assumption (5) on L in (3) might look a bit strange. But for our proof of the a priori bounds in Hg it is needed. For a discussion of this assumption see Section 6. It is not the main concern of this paper but whenever one can show an existence theorem the question of uniqueness arises. In Section 4.2 a

simple condition is given when uniqueness holds. Section 7 is devoted to Robin boundary conditions.

3. Schaefer's fixed point theorem. The proof of Theorem 4 is mainly based on Schaefer's fixed point theorem (cf. [17]). We do not apply the original formulation of Schaefer's theorem from his paper, but a slightly different version. Actually it does not matter which version we apply but we prefer the version presented below.

Theorem 5 (Schaefer's fixed point theorem). Let X be a locally convex Hausdorff space and T : X ^ X a continuous mapping. Let

S = {u G X|30 < t < 1 : u = tTu}.

Let ||-|| be a continuous semi-norm on X, let p > 0 be a number and Kp = {u G X : ||u|| <

P}. If

(i) SC Kp and

(ii) TKp C X is relatively compact,

then T has a fixed point u = Tu.

A proof which is based on the Tychonoff—Schauder fixed point theorem (cf. [18]) can be found in [2, Section 2]. Actually we could replace (i) by the more general condition u G S ^ ||u|| = p. This shows the connection of Schaefer's theorem to the well known degree theory of Leray—Schauder.

4. A priori bounds and domination. We consider the class of boundary value problems (10) and prove the Theorems 1, 2 and 3. It is easy to generalize these Theorems also to sub- and supersolutions.

4-1- A priori bounds in Hq(Q). Proof of Theorem 1. Let u be a weak solution of (10). We choose u G Hq(Q) as a test function for (10). Since A1 + m — e > 0, |s| > |<r(x, s)| and the fact that <r(x, s) and s have the same sign for every s G R and almost all x G Q by the coerciveness condition (3) we may deduce

||Vu||

L2(Q) + M ||u|L2(n) < t(A1 + M — e) ||u|L2(w) +

+ tL HVu||L2(.) ||u||L2(u0 + t 11 h|L2(Q) ||u||L2(o0 . (11)

Let e1,e2 > 0 with e = e1 + e2 for which there is a ¿1 G (0,1] such that e1 = ¿1A1. If A1 =0 we choose ¿1 = 1. Appropriately inserting the Poincare—Steklov inequality into (11) yields

((1 — ¿1)A1 + M) ||u||L2m + ¿1 | Vu|L2(w) < t((1 — ¿1)A1 + M — e2) |u|L2(w) +

+ tL ||Vu|

L2(U) ||u|L2(w) + t ||h|L2(Q) ||u|L2(w) . After rearranging the terms and dividing by t we get

((t-1 — 1)(A1 — e1 + m) + e2) |u|L2(w) + t-1¿l |Vu|L2(w) <

< L ||Vu||

L2(W) ||u|L2(w) + ||h|L2(Q) ||u|L2(w) .

£2 \\urw + 5, \\Vurw < \\Vurw + — \\urw + ||fe||L2(n) \\u\\w (12)

Since (t 1 — 1)(Ai — £i + y) > 0 and t 1 ¿q > ¿q we deduce with the help of the inequality of arithmetic and geometric mean

r2

12 , r iivt I|2 ^ i- IIVT ii2 , II n2

5i

i. e.

4^2 < L2 +4Ji yhyLa(n) .

If we set £1 = min{e/2, A1} we easily see that 4£1e2 > L2 and thus we deduce that there exists a p0 > 0 such that ||w||i2(w) < p0 ||h||i2(q). Inserting this in (11) we get the desired a priori bound. □

4-2. Uniqueness. A second view on the proof of the Theorem 1 gives rise to a uniqueness result for (2) if we assume the monotonicity condition

(f (x, S2, 6) — f (x, S1, ¿1 ))(S2 — S1) < (A1 — e)(s2 — si)2 + L |6 — 1 h — . (13)

Proposition 6. Let f satisfy the monotonicity condition (13). Then the Dirichlet problem (2) has at most one weak solution.

Proof. Let us assume that there exist two solutions u1 and u2. Then u = u2 — u1 G

Hq(9) can serve as a test function for (2) with u replaced by u1 or u2. Then (13) leads to

2 2 (*)

| Vu|L2(n) < (Ai — e) |u|L2(n) + L | Vu|i2(n) ||u|i2(n) <

(*) 2

< |Vu|L2(n) + (L — Lmax(e, Ai)) || VuH^n) |M|L2(n) •

We deduce u = 0. The second inequality (*) follows from the Poincare—Steklov inequality and

LmBX(s, ao ||Vu\\L2(n) ||M||L2(n) < ||-SJu\\l2(.) + £maX4^Al)2 IMILm

for ¿q = e/(2AQ) if e < 2Aq and ¿q = 1 else. □

4-3. A priori bounds in Lq**(9). Proof of Theorem 2. Theorem 1 already implies an a priori bound for the L2* (9)-norm by the Sobolev embedding theorem. With the help of Moser's iteration method we also achieve bounds with respect to higher order Lebesgue norms. We learned the Moser iteration technique from the proof of [6, Theorem 8.15] and could generalize it to our situation. For equations of the form —Au = g G Lq(9) on bounded domains and Robin boundary conditions see also [9, Theorem 4.1].

(i) Making Moser iteration possible. For 2 < r < to and 2 < p < q we define

M(r)=max{||u||L2(n), ||u||Lr(n)} and p(p) = M(p) + ||h||Lp(n) •

By Lyapunov's interpolation inequality1 for Lebesgue spaces we see that M is an increasing function. Moreover M : [2, to] ^ [0, to] is continuous. A standard test function argument shows that the coerciveness condition (3) implies

M(2*^)2^ < Cq^2p(p)M(p'(2£ — 1))2^-1 (14)

1 Lyapunov's inequality [10, Section 2.9]: ||v||_Lr(q) < ||v||L (q) ||v||Lq(n) for all measurable functions

v if 1/r = (1 — 0)/p + 0/q where 0 < r,p,q < <x and 0 < 0 < 1.

for all fl > 1 and 2 < p < q. Here 2* is equal to 2d/(d — 2) if d > 2, or some sufficiently big number else. The constant Cq > 1 only depends on d, S and L.

For the proof of (14) one can make the same ansatz as in the proof of [6, Theorem 8.15] with k = 0. That is, we use a test function which is proportional to |u|2^-2 u for small values |u(x)| and proportional to u for big values of |u(x)|. The crucial point in the proof is the validity of the Sobolev embedding Hg(Q) ^ L2* (Q).

(ii) Moser iteration. We distinguish the two cases 2q < d and 2q > d. Case 2q < d: For the beginning let us assume that 2 < p < max{2*, q}. By Sobolev's embedding theorem p(p) < to. Note that 2*fl = p'(2fl — 1) is equivalent to 2*fl = p**. Thus (14) implies M(p**) < to and therefore

/ * * \ 2

M(p") < C\ ^j (m(p) + \\h\\Lp{Q)) . (15)

This yields p(p**) < to. By iterating the preceding argument we get M(p**) < to for all 2 < p < q and a uniform estimate holds. By the continuity of M this is also true for p = q. By Lyapunov's inequality at the cost of a bigger constant we can replace M(q) on the right hand side of (15) by ||u||i2(fi) and the claim follows.

Case 2q > d: From the first case we already know that M(r) < to for all 2 < r < to. Thus p(q) < to. Let us recursively define the increasing sequence (fln) by fl0 = 1,fln = \ + xPn-i where x = ^r > 1- Then we deduce from (14) with p = q that

M(2*fln) < fl^"(Cip(q))1/(2^n}M(2*fl„-i)1-1/(2^n).

Observe that 1 — = • This gives immediately

/ N X 1/(^n)

N — m\, , ,,n /a , n

m(2*flN) < n flmm "m (Cip(q))1-x M(2*)

*)XN/^n

C(N,d,q)

It is a simple exercise to show that

(a) xN/flN converges from above to 0 = (2x — 2)/(2x — 1) G (0,1) and

(b) there exists a constant C(d, q) > 0 such that C(N, d, q) < C(d, q). Thus

M(2*flN) < C2p(q)1-xN/^NM(2*)xN/^n.

The constant C2 depends on d, S, L and q. Since M is continuous we may let N tend to infinity and deduce from (b) that

M(to) < C2p(q)1-0M(2*)0

for some 0 = 0(d, q) G (0,1). Since M(2*) < M(to) we may simply divide by M(to)0 to get M(to) < C3p(q). Now the claim follows as in the case 2q < d. □

Domination in Lq« (Q). Proof of Theorem 3. We only show the assertion about the Dirichlet problem (9). The statement about (8) is proved similarly.

1

Let us define the operator A with domain {v G Hq(Q) : Av G L2(Q)} which acts as

—A — Ai + £. By the Poincare—Steklov inequality this operator is invertible.

Furthermore let us define the nonlinear but continuous operator S : Hq(Q) ^ L2(Q) by Sv = L |Vv| + h. Then (9) is equivalent to the fixed point problem

v = A-iSv =: Tv

for the operator T : ffQ(Q) ^ Hq(Q). We apply Banach's contraction mapping principle to show existence and uniqueness of a solution. It is important to choose an appropriate norm on Hq(Q) which makes T a contraction mapping.

Let vq, v2 G Hq(Q) be arbitrary, m := Tvj for i = 1, 2 and v := v2 —vQ and u := u2 — mq. Then

(AM, = (Sv2 — SVq, V^

Let £q,£2 > 0 with £ = £Q + £2 for which there is a G (0,1] such that £Q = ¿iAi. If Aq =0 we choose = 1. From the last equation with ^ = u e Hq(Q) follows

Г2

2 2 2 L 2

S1 IIVmIIЬ2(П) + £2 IMIl2(S1) < «¿1 ||V«||i2(n) + ||M||i2(n)

for all 0 < a < 1 as in the derivation of (12) in the proof of Theorem 1. If we set £i = min{£/2, Ai} again we see that 4^£2 > L2 and thus there exists an a, maybe close

L 2

to 1, such that £ := £2 — > 0. Thus we proved that T is a contraction mapping with

( i i \ contraction constant л/а with respect to the norm (¿i || VwH^^ + e' |M|L2(q) J • This

shows that the Dirichlet problem (9) has a unique solution v.

It remains to show v > 0 and и < v for every solution и of (10). We only show the more difficult second assertion. Therefore let w = (и — гТ)+ G Hq(Q) and g = (Ai + ц — e)v + L |W| + h > 0. Let w serve as a test function for

—A(u — v) + fj,(u — v) <tba(x,u,Vu)xu(x) — g(x) (x G Cl), u{x)-v{x) <0 (xGdQ).

It follows that

IIVwHjL2(o) + м INlLL2(n) < t Vm+) - <

J ш

< t (A1 + м - £)w2 + L |Vw| < (A1 + M - £) ||w|LL2(n) + L 1 Vw|L2(n) I|w|L2(n) •

J ш

As in the proof of Proposition 6 we deduce w = 0. This means и < v. □

5. Proof of the main theorem. Now we prove Theorem 4. Observe that the conditions (3) and (4) in conjunction with the Theorems 2 and 3 imply

\f(x,v(x),0\ </o(x) + Lo|£| V v<v<v, £ G (16)

for some function f G L2(Q), f > 0.

We divide the proof into three steps. In the first step we do not consider the actual Dirichlet problem, but a truncated version of it. This truncation procedure makes it

possible to apply Schaefer's fixed point theorem in the locally convex space HQoc(9) to achieve the existence of a sequence of weak solutions of such truncated problems. This will be the second step. In the last step we show that a subsequence converges to a weak solution of (2). Such a pattern was applied in [11] to get an existence theorem for a parabolic equation on an arbitrary open set.

(i) Truncation. It is well known that there is an increasing sequence of open sets (Qk) with C^-boundary such that is compact and included in 9fc+i and 9, and such that their union is 9. Let a : 9 x [R —> [R be as in (10) but with vq = v and v\ = v. For y > 0 such that Aq + y — e > 0 let us define the Caratheodory function as in (10) and b(x, s, = f (x, s, + ys. For v G HQoc(9) we consider the following truncated Dirichlet problem

—Au + yu = 6CT(x,v, Vv)xnk (x) (x G 9), ( )

u(x) =0 (x G d9). ( )

(ii) Schaefer's fixed point argument. From (16) we deduce that the Nemytskii operator

v ^ (x, v, Vv)xnk is continuous from HQoc(9) to L2(9) and maps ||-||Hi(n )-bounded2 £

operator

sets of HQoc(9) into bounded sets in L2(9) (cf. [12, Theorem 19.2]). Thus (17) defines an

Tfc : HQoc(9) ^ HQ n H2oc(9) ^ HQoc(9) by Tfcv = u.

It is a continuous mapping on the locally convex space HQoc(9) and satisfies condition (ii) in Theorem 5 for X = HQoc(9), ||-|| = ||-||hi(fifc) and every p > 0. Let

Sfc = {u G HQoc(9) : u = tTfcu for some 0 < t < 1}

be the Schaefer set with respect to Tk. If we can show that is ||-||hi(fifc)-bounded then we get the existence of at least one fixed point uk = Tkuk for every k by Theorem 5. By Theorem 1 an even stronger assertion is true:

u G ^ ||u|H0i (n) < p>

for some constant p which does not depend on k. Therefore we get the existence of an HQ(9)-bounded sequence (uk) of fixed points, as desired.

(iii) Convergence to a solution. By passing to a subsequence if necessary, we may assume that uk converges weakly to some function u in HQ(9). Furthermore we may assume that this convergence is also true in the pointwise sense (almost everywhere), since the embedding if/oc(9) > L2j;oc(9) is compact. By Theorem 3 we know that v<Uk<v for all k and therefore these functions satisfy

—Aufc + yufc = 6(x,ufc, Vufc)xnfc (x) (x G 9), ( )

uk(x) =0 (x G d9). (18)

By (16) the estimate ||b(-, uk, Vuk(n) < ||f0||l(n) + yP+L0p is true for all k. Thus (18) implies that (uk) is also bounded with values in the domain of the Dirichlet—Laplacian which is embedded into H2oc(9). Thus we may assume that (uk) also strongly converges

2 A subset of Hloc(Q) is called H-Hn^n^-bounded if it is included in the ball {u € Hloc(H) |M|ffi(nfc) < r} for some r > 0.

in HQoc(9) to u. Again by [12, Theorem 19.2] this implies that b(x, uk, Vuk) strongly converges in L2jl oc(9) to b(x, u, Vu) G L2(9). The arguments above allow us to take the limit k —>■ to in (18) which shows that u is a weak solution of (2). □

6. Can the bound for L be improved? It is clear that in general we lose the existence of solutions for (2) if e in condition (3) is allowed to be zero. Now we ask

• Do we also lose the existence of solutions if L > Lmax(e, A1)7

Unfortunately we do not know the answer of this question. In some standard situations it is not difficult to adopt the proof of Theorem 1 (which exclusively uses that L is bounded from above) to get existence of solutions. For example if f (x, s, £) is of the form f1 (x, s) — b(x) • £ for some C1 vector field b with V • b < 0, then one can prove a priori bounds in Hq(9) in the sense of Theorem 1 where C does not depend on b. This is due to 2 fn b • (Vu)udx > fn V • (bu2)dx = 0 if u G Hq(9). We remark that the structure condition V • b < 0 arises in applications (see [13] and references therein).

The existence theorem in section 2.2 is stated only for real valued boundary value problems. The only reason why we cannot extend it to elliptic complex valued problems (or even systems) is that Theorem 3 (a priori domination) does not extend to this situation. However, if we strengthen (7), an existence theorem can be formulated for complex problems. Keeping this in mind we now consider a complex valued problem

J —Au + ib • Vu — (Aq — e)u = g(x) (x G 9), ( )

\ u(x) =0 (x G d9), (19)

where b G is a constant vector, g G L2(9) and show

Theorem 7. Let r > Lmax(e, Aq) and d > 2. Then there exists b G with |b| = r an open set 9 and g G L2(9) such that (19) has no solutions.

Proof. It is convenient to distinguish three cases: (i) Aq = 0, (ii) e < 2Aq and (iii) 0 < 2Ai < e.

Case (iii): 0 < 2Ai < e. We choose 9 = (0, x We write x = (xi, x') G 9

and similarly b = (bQ, b'). After taking Fourier transform

u(Z, £)= / e-iXlZu(xQ, x')dxQ + u(xQ,x')dx'

Jo Jpd-1

for ( G a/XiZ and £ G we see that (19) is equivalent to

(c2 + lei2 - &ic - b' ■ e - (Ai - e)) a = g g i2 {y/xa) l^-1), (20)

V-V-'

where a solution u must necessarily belong to /2(VXj"Z) <8>2 L2([Rd_1). It is possible to find b G such that

61 = 2^ and |6| = r. (21)

Thus 6/2 G a/AiZ and p(b/2) < 0. Therefore we can find ^ G such that p{yj£) = 0, and it is possible to find g such that (20) has no solution. As a consequence (19) has no solutions for the corresponding g.

Case (i): Ai = 0. When we interpret l from the third case as then the above argumentation also works in this case if we choose b with |5| = r > 2y/e arbitrary and note that p{y/eb/ |6|) < 0.

Case (ii): e < 2\\. Instead of (21) we choose b with b\ = ej\f\[ and |5| = r > ej\f\[ arbitrary and note that p(a/X¡", 0) = 0. □

Remark 8. We only used the fact that a weak solution of (19) has to be in H 1(l) and satisfies the equation in the sense of distributions. Therefore this example works not only for Dirichlet boundary conditions.

7. Robin boundary conditions. We can generalize the problem (2) by considering more general boundary conditions than merely Dirichlet boundary conditions. In the following we treat Robin boundary conditions, which can be defined on arbitrary open sets (cf. [9, 14, 15]). As a result of this section it turns out that a version of Theorem 4 remains true for Robin boundary conditions under a strict positivity assumption. For general Robin boundary condition we need an additional assumption (cf. (24)) which replaces the Sobolev embedding theorem which is necessary to establish the a priori bounds in the Lebesgue spaces by Moser iteration.

7.1. Definition of generalized Robin boundary conditions. Let ^ be a (positive) measure on the Borel a-algebra of dl. We define the positive form

a(u, v) = / VuVvdx + / uvd^, (22)

Jn Jen

with domain D(a) = {u G H1(Q) fl Cc(n)| fdQ |w|2 djjb < oo}. We write a(u) := a(u,u) and equip D(a) with the norm (a(u) + ||w||L2(Q))1/2. If a is closable (i.e. the completion of D(a) embeds injectively into L2(l)) we may define a self-adjoint operator by

D(A) = {«£ V\3g G L2(Q)Vv G V : a(u, v) = (g, v) L2{n)} , Au = -g,

where a denotes the closure of a. This operator acts as Au = Au G ¿2(0). Thus D(A) C H2oc(l), i. e. local maximal regularity does not depend on the specific boundary conditions (see [6, Theorem 8.8]). Even if a is not closable there exists a (unique) maximal closable positive form ar smaller than a in the following sense: The form ar is smaller than a, that is D(a) C D(ar) and ar(u) < a(u) for every u G D(a), and every closable positive form b which is smaller than a is also smaller than ar (see [16, Supplementary material, Theorem

S. 15]).

In [15] the authors give a characterization of ar by means of the relative capacity with respect to iX The relative capacity for a (not necessarily Borel-) subset A C Q is defined by

Cap^(A) = inf | J (JV«|2 + M2) dx | u G H1^), 30Cfi relatively open :

A C O and u > 1 a.e. on o|

Here H1(Q) denotes the closure of H1(Q) fl C(Q) in H1(Q). The relative capacity is an outer measure. A property is said to hold relatively quasi-everywhere (r.q.e.) if it holds

on Q\N where N is a set with Cap^(N) = 0. Every u G H1(Q) has a r.q.e. unique relatively quasi-continuous representative which we denote by u. This is a function u = u a.e. on Q such that for each e > 0 there exists a relatively open subset uj C Q with Cap-^(Q\i>j) < e and u restricted to uj is continuous (see [17, Chapter I, Theorem 8.2.1]). Let rM = {x G d9|3r > 0 : y({y G d9| |x — y| < r}) < to} be the maximal open set where y is locally finite. For a Borel set S C rM we define as to be the positive form with domain D(as) = D(a) which is given by (22) where the boundary integral over d9 is replaced by the same integral over S. Note that as is smaller then a. The Borel set S C is called y-admissible if Cap^(A) = 0 implies y(A) = 0 for each Borel-subset A of S.

Theorem 9 ([14, 15]). There exists a y-admissible set S C such that Cap^iT^S) = 0 and as = ar. Moreover

D(as) = G ii1(9)|m = 0 r.q.e on dil\S and J |w|2 djj, < to

as(u, v) = / VuVvdx + uvdy. JQ JS

S is unique up to a y-null set.

In [15, Example 4.3] the authors constructed a bounded domain 9 such that d9 is not admissible for the (d — 1)-dimensional Hausdorff-measure a although it has finite measure with respect to a. This shows that the maximal admissible set S given by the above Theorem does not in general coincide with rM.

7.2. A generalized existence theorem. Let y be a measure on the Borel a-algebra of d9 and S the maximal y-admissible set from Theorem 9. We consider the boundary value problem

—Au(x) = f (x,u(x), Vu(x)) (x G 9),

u(x) =0 (x G d9\S), (23)

+ u(x)dfj,(x) =0 (x G S).

We call u a weak solution of this problem iff u G D(as) and /(x, u, Vw) G ¿2(9) such that

du /

as(u,ip)= / f(x,u,'S7u)ipdx \/<f G D(as). JQ

m

Let 2 < d < oo be a real number. We assume that

All conditions on f remain unchanged as compared to Section 2 but (7) has to be satisfied if and only if q < d/2, where we redefine q** = qci/(cZ — 2q)+. The (for simplicity) excluded value for q is d/2 instead of d/2. Furthermore, Aq g [0, to) is the optimal constant such that Ai |M|^2(q) < as(u) is true for all u G D(as)•

Theorem 10. Under the above conditions the boundary value problem (23) has at least one weak solution.

It is not difficult to generalize Theorems 1, 2 and 3 and the argumentation in Section 5 to this more general setting. In fact, (24) replaces the Sobolev-embedding Hq (l) ^ L2* (l) which has to be used in step (i) of the proof of Theorem 2 and the following lemma guarantees that the test function argument described there also works in our more general situation:

Lemma 11. Let P > 1 and t0 > 0. Let H(s) = |s|^-1 s for all real |s| < t o. Extend H affine linearly to a C1 -function. Then

u G D(as) ==>■ H{u) G D(as) and H{u) = H{u)~

where U and H(u)~ denote the relatively quasi-continuous representatives of u and H(u) respectively.

Proof. Essentially we only have to proof that H(u) is the relatively quasi-continuous

representative of H(u). But this is easy since if u is continuous on some set then so is H(u). □

It is only left to find examples where (24) is true for some d. Of course, if dl satisfies a uniform lipschitz condition then it is true for d = d by the Sobolev embedding theorem. But there are also other situations where (24) is true. For this purpose set d^ = P(x)da where a is the (d — 1)-dimensional Hausdorff-measure restricted to dl. The Borel function P is bounded from below, that is P(x) > p0 > 0 for some constant p0. Then it is a consequence of an inequality due to Maz'ya that (24) holds with d = 2d. We refer to [15, Chapter 5] for the short proof.

Remark 12. One could ask if the a priori bounds derived from (24) by Moser iteration are the best possible. Indeed they are: [9, Theorem 5.11].

Acknowledgments. I am most grateful to Ralph Chill and Alexander I. Nazarov who helped me with their advice and knowledge. I am also grateful to Arina A. Arkhipova for useful comments improving the final version of the article.

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Статья поступила в редакцию 26 марта 2015 г.

Сведения об авторе

Reinhard Stahn — PhD student; [email protected]