Boundary regularity for elliptic problems with continuous coefficients

Transcription

1 Boundary regularity for ellitic roblems with continuous coefficients Lisa Beck Abstract: We consider weak solutions of second order nonlinear ellitic systems in divergence form or of quasi-convex variational integrals with continuous coefficients under suerquadratic growth conditions. Via the method of A-harmonic aroximation we give a characterization of regular boundary oints using and extending some new techniques recently develoed by M. Foss & G. Mingione in 15]. 1 Introduction and results In this aer we resent a characterization of regular boundary oints in the regularity theory of vectorial ellitic and variational roblems by extending the techniques and the results of Foss & Mingione in 15] to the boundary. We first consider weak solutions u W 1, (Ω, R N ) of a general homogeneous system of second order ellitic equations in divergence form div a(, u, Du) = 0 in Ω, (1.1) where Ω is a bounded domain in R n and a: Ω R N R nn R nn is a continuous vector field on which we imose standard boundedness, differentiability, growth and elliticity conditions: z a(,, z) is of class C 1, and for fixed 0 < ν L and all x, x Ω, u, ū R N, and z, z, λ R nn there holds: a(x, u, z) Dz a(x, u, z) ( 1 z ) L ( 1 z ) 1, D z a(x, u, z) λ λ ν ( 1 z ) λ, ( ) a(x, u, z) a( x, ū, z) 1 ( L 1 z ω x x u ū ) (1.). Dz a(x, u, z) D z a(x, u, z) ( L µ z z ) ( ) 1 z z. 1 z z Here n, N,, and µ, ω : R R are two moduli of continuity, i. e. bounded by 1 (without loss of generality), concave and non-decreasing such that lim 0 ω() = 0 = lim 0 µ(). The role of the modulus of continuity ω( ) will be the crucial oint in our aer; we remark that, in the sequel, we confine ourselves to the vectorial case. For the scalar case we refer to 15] and the references therein. If we assume a Hölder condition of the form ω(t) t α for some α (0, 1), t R, i. e., (1 z ) 1 a(x, u, z) is Hölder continuous in the variables (x, u) uniformly with resect to z, then it is known (see 17]; 14] for the variational case) that standard growth and elliticity assumtions on the coefficients imly artially Hölder continuous first derivatives of the weak solution u, which means Hölder continuity outside the singular set of Lebesgue measure 0, with otimal Hölder exonent α. Moreover, assuming that the boundary data are sufficiently smooth, general criteria for Du to be regular in a neighbourhood of a given boundary oint were obtained by Grotowski and Hamburger (see 0, 1]) using boundary versions of the method of A-harmonic aroximation and of the blow-u technique, resectively. The assumtion on ω( ) was weakened to Dini-continuous coefficients, where r 0 ω() d < is fulfilled for some r > 0, which still allows to conclude a artial regularity result for Du (see 8, 7]; 9] for the variational case). Moreover, a condition of the form lim su 0 ω() log( 1 ) = 0 ensures in the case of variational functionals under non-standard growth without u-deendency (see 3], Theorem.1) to infer u C 0,α loc (Ω, RN ) for every α (0, 1). Assuming merely the continuity of the coefficients with resect to the variable (x, u) without any further structural assumtions, Camanato roved low order artial regularity in 5], namely that the weak L. Beck, Mathematisches Institut der Friedrich-Alexander-Universität Erlangen-Nürnberg, Bismarckstr. 1 1/, Erlangen, Germany. beck@mi.uni-erlangen.de 1

2 L. Beck solution u is Hölder continuous with every exonent α (0, 1) outside a negligible closed subset of Ω, for the low dimensional case, where n (cf. 6] for similar estimates u to the boundary; see also 3] for the variational setting). Moreover, the Hausdorff dimension of the singular set is bounded by n from above imlying that actually almost every boundary oint is verified to be a regular one. In contrast, in the case of quasi-convex variational integrals, these methods do not aly, and a similar low dimensional result was obtained only under the assumtion ω(t) t α (cf. 3], Theorem 1.5). However, for general dimensions, the question of low order artial regularity under a continuity assumtion remained unsolved for a long time, until Foss & Mingione gave a ositive answer in 15] both for weak solutions of ellitic systems and for local minimizers of quasi-convex variational integrals. The aim of our aer is now to extend the characterization of regular oints u to the boundary. For this urose we denote by Reg Ω u the regular boundary oints of u in the sense that Reg Ω u := { x 0 Ω: u C 0,α (U(x 0 ) Ω, R N ) for every α (0, 1) and some neighbourhood U(x 0 ) of x 0 }, and the set of singular boundary oints by Sing Ω u := Ω \ Reg Ω u. Analogously for fixed α (0, 1) we define Reg Ω,α u := { x 0 : u C 0,α (U(x 0 ) Ω, R N ) for some neighbourhood U(x 0 ) of x 0 } and Sing Ω,α u := Ω \ Reg Ω,α u. Our first theorem then rovides a characterization of the regular boundary oints analogous to the characterization of regular oints in the interior of Ω (see 15], Theorem 1.1): Theorem 1.1: Consider, Ω R n, n, a bounded domain of class C 1 and a ma g C 1 ( Ω, R N ). Let u W 1, (Ω, R N ) be a weak solution of system (1.1) under the assumtions (1.) with boundary values u = g on Ω. Then there holds: { Sing Ω u x 0 Ω: lim inf 0 or lim inf 0 Ω B (x 0) β D ν Ω (x 0)u (D ν Ω (x 0)u) Ω B(x 0) (1 (D ν Ω (x 0)u) Ω B(x 0) ) > 0 D ν Ω (x 0)u > 0 Ω B (x 0) for every β (0, ); here ν Ω (x 0 ) denotes the inward-ointing unit normal vector to Ω in x 0. Moreover, for every α (0, 1) there exists s > 0 deending only on n, N,, ν, L, α, β, Ω, g, ω( ) and µ( ) such that the following inclusion holds for every β (0, ): { Sing Ω,α u x 0 Ω: lim inf 0 or lim inf 0 Ω B (x 0) β D ν Ω (x 0)u (D ν Ω (x 0)u) x0, (1 (D ν Ω (x 0)u) Ω B(x 0) ) s } D ν Ω (x 0)u s. Ω B (x 0) We note that for general dimensions the roblem of knowing whether there might exist regular boundary oints, even in the case of Hölder continuous coefficients with exonent α < 1, remains oen (cf. 1]), unless we have some additional structural condition (as e. g. a slitting condition, see ] for minima). In the second art of the aer we consider variational integrals of the form Fu] := F (x, u, Du), (1.3) Ω where the integrand F : Ω R N R nn R is strictly quasi-convex, continuous and grows olynomially. More recisely, we assume that z F (,, z) is of class C and that F satisfies for fixed 0 < ν L and all x, x Ω, u, ū R N, and z, z R nn the following assumtions: ν (1 z ) F (x, u, z) L (1 z ) ν (1 z Dϕ(y) ) Dϕ(y) dy ] (0,1) n (0,1) F (x, u, z Dϕ(y)) F (x, u, z) dy n F (x, u, z) F ( x, ū, z) L ( 1 z ) ( ω x x u ū ) (1.4), D zz F (x, u, z) D zz F (x, u, z) L µ ( z z ) ( ) 1 z z. 1 z z }

3 Boundary regularity for ellitic roblems 3 The functions µ( ) and ω( ) are those already considered in the ellitic case, and for (1.4), which is called strict quasi-convexity condition, we assume ϕ C 0 ((0, 1) n, R N ). We note here that quasi-convexity is an extension of convexity to a global roerty and is essentially equivalent to lower semicontinuity (cf. 1]). Alying ste of age 6 in 5], we may also assume a growth condition on the first derivatives of the form D z f(x, u, z) L(1 z ) 1. Moreover, it can be verified that the conditions (1.4) above (see 6], Theorem 4.3) imly the strict elliticity of the matrix D f in the sense of Legendre-Hadamard, and therefore we may also assume for all ξ R N, η R n. ν (1 z ) ξ η D zz F (x, u, z) ξ η ξ η L (1 z ) ξ η We note that in the case of Hölder continuity of coefficients a artial regularity theory for the gradient of minimizers has been established in the by now classical aers 14,, 7], while artial Hölder continuity in the interior in the case of general continuous coefficients has been again roved in 15]. Our second theorem now yields a characterization of the regular boundary oints of minimizers of quasiconvex integrals corresonding to the ellitic case: Theorem 1.: Consider, Ω R n, n, a bounded domain of class C 1 and a ma g C 1 ( Ω, R N ). Let u W 1, (Ω, R N ) be a local minimizer of the functional F ] in (1.3) under the assumtions (1.4) with boundary values u = g on Ω. Then there holds: { D ν Ω (x Sing Ω u x 0 Ω: lim inf 0)u (D ν Ω (x 0)u) Ω B(x 0) 0 Ω B (x 0) (1 (D ν Ω (x 0)u) Ω B(x 0) ) > 0 } or lim inf 0 β D ν Ω (x 0)u > 0 Ω B (x 0) for every β (0, ). Moreover, for every α (0, 1) there exists s > 0 deending only on n, N,, ν, L, α, β, Ω, g, ω( ) and µ( ) such that for every β (0, ) the following inclusion holds: { D ν Ω (x Sing Ω,α u x 0 Ω: lim inf 0)u (D ν Ω (x 0)u) Ω B(x 0) 0 Ω B (x 0) (1 (D ν Ω (x 0)u) Ω B(x 0) ) s } or lim inf 0 β D ν Ω (x 0)u s Ω B (x 0) Finally we briefly comment on the techniques used in the roofs and on the modifications necessary to handle the boundary situation: regularity roofs for both nonlinear systems and functionals are usually based on a comarison rincile in order to establish an excess decay estimate. In the resent situation, the excess quantity introduced by Foss & Mingione consists of three terms: the first involving the averaged mean deviation of the derivative of the weak solution (re-normalized by the factor (1 (D ν Ω (x 0)u) Ω B(x 0) ) which might diverge due to the fact that we cannot exect to obtain Lischitz estimates even if the boundary data are smooth), the second involving the radius of the ball B (x 0 ) and finally the Morrey-tye excess M(x 0, ) quantifying the oscillations of the weak solution u. The aroriate decay of this excess quantity is now obtained by a linearization argument (combined with the Ekeland variational rincile in the case of variational integrals), namely by freezing the coefficients and the functional, resectively, in order to obtain an ellitic system A with constant coefficients. In the second ste, the comarison with an A-harmonic ma (for which good a riori estimates are available u to the boundary) is made ossible by the technique of A-harmonic aroximation (for details we refer to 10] and, for a more general form in the setting of geometric measure theory 13]). Our main goal, the Hölder continuity of u on a relative neighbourhood of a given boundary oint x 0 where the excess is small, can then be established in the model case of the uer half unit ball (which is sufficient for the general situation) by roceeding similarly to existing aers concerned with boundary regularity (see e. g. 0]); here, we mention that our boundary excess describes only the behaviour of the normal derivative of u but aroriate boundary versions of the Cacciooli and the Poincaré inequality allow us to control the full derivative of u. Hence, the excess at the boundary is now used to get control over the corresonding excess quantity on balls in the interior within a neighbourhood of x 0, and a combination with the interior case then yields the desired regularity result. In the sequel, we set our main focus on the treatment of the boundary situation, but, as indicated in 15], the methods also aly to cover inhomogeneous ellitic system and almost minimizers of integral functionals..

4 4 L. Beck Preliminaries We start with some remarks on the notation used below: we write B (x 0 ) = { x R n : x x 0 < } and B (x 0 ) = { x R n : x n > 0, x x 0 < } for a ball or an uer half-ball, resectively, centred on a oint x 0 ( R n1 {0} in the latter case) with radius > 0. Sometimes it will be convenient to treat the n-th comonent of x R n searately; therefore, we set x = (x, x n ) where x = (x 1,..., x n1 ). Furthermore, we write (x 0 ) = { x R n : x x 0 <, x n = 0 }, for x 0 R n1 {0}. In the case x 0 = 0 we set B := B (0), B := B 1 as well as B := B (0), B := B 1 with := (0), := 1. We also introduce the following notation for W 1, -functions defined on some half-ball B (x 0 ) and which vanish (in the sense of traces) on the flat art of the boundary: W 1, (B (x 0 ), R N ) := { u W 1, (B (x 0 ), R N ) : u = 0 on (x 0 ) }. Let L n denote the n-dimensional Lebesgue measure. For any bounded, measurable set X R n with L n (X) =: X > 0, we denote the mean value of a function h L 1 (X, R N ) by (h) X = X h, and, in articular, we use the abbreviation (h) x0, for the mean value on B (x 0 ) or on B (x 0 ), resectively. The constants c aearing in the different estimates will all be chosen greater than or equal to 1, and they may vary from line to line. We consider a bounded domain Ω in R n, for some n. The boundary of Ω is assumed to be of class C 1 with modulus of continuity τ( ); this means that for every oint x 0 Ω there exist a radius r > 0 and a function h : R n1 R of class C 1 such that (u to an isometry) Ω is locally reresented by Ω B r (x 0 ) = { x B r (x 0 ) : x n > h(x ) }. Thus we can locally straighten the boundary Ω via a C 1 -transformation T defined by T(x, x n ) = ( x, x n h(x ) ). We recall that u is a weak solution of (1.1) with boundary values g under the assumtions (1.) if u is a W 1, (Ω, R N )-ma such that a(x, u, Du) Dϕ = 0 for every ϕ W 1, 0 (Ω, R N ) Ω and if u = g on Ω in the sense of traces. Further, u is a local minimizers of the functional F ] with boundary values g under the assumtions (1.4) if u is a W 1, (Ω, R N )-ma such that and if u = g on Ω in the sense of traces. Fu] Fv] for every v u W 1, 0 (Ω, R N ) Firstly we recall a boundary version of Poincaré s inequality W 1, (B R, RN )-mas. The fact that u vanishes on allows to estimate the integral over u by the integral of the normal derivative D n u only rather than the full derivative. Lemma.1 (4], Lemma 3.4): For functions u W 1, (B R (x 0), R N ) with x 0 R n1 {0} there holds: u R D n u. B R (x0) B R (x0) We next want to state results from the linear theory. Firstly, we recall the following u-to-the-boundary version of the A-harmonic aroximation lemma: Lemma. (19], Lemma.4; 0], Lemma.3.): Consider fixed ositive ν and L, and n, N N with n. Then for any given ε > 0 there exists δ = δ(n, N, λ, L, ε) (0, 1] with the following roerty: if A is a bilinear form on R nn satisfying ν ξ η A (ξ η, ξ η) L ξ η (.1) for all ξ R N, η R n, and if w W 1, (B (x 0 ), R N ) (for some > 0, x 0 R n1 {0}) with B (x Dw 0) 1 is aroximately A-harmonic in the sense that A (Dw, Dϕ) δ su Dϕ B (x 0) B (x 0)

5 Boundary regularity for ellitic roblems 5 for all ϕ C0(B 1 (x 0 ), R N ), then there exists an A-harmonic function h W 1, (B (x 0 ), R N ) such that B (x 0) Dh 1 and w h ε. B (x 0) That A-harmonic mas are indeed smooth, is the statement of the next lemma: Lemma.3 (19], Theorem.3): Consider fixed ositive ν and L, and n, N N with n. Then there exists a constant c h deending only on n, N, L and ν such that for every bilinear form A on R nn with uer bound L and elliticity constant ν and any A harmonic ma h W 1, (B (x 0 ), R N ) (for some > 0, x 0 R n1 {0}) there holds: su Dh su D h c h Dh. B / (x0) B / (x0) B (x 0) Given a functions u L (B (x 0 ), R N ) we denote by P x 0, the unique function minimizing the functional P B (x 0) u P amongst all functions P of the form P (x) = Q x n for some Q R N. P x 0, = Q x 0, x n is then given via for c Q = ( B 1 4], Lemma : Q x 0, := c Q (n) u(x) x n B (x 0) x n ) 1 = n. The following lemma rovides exlicit estimates for Q x 0, similar to Lemma.4: Let u W 1, (B (x 0 ), R N ) for some x 0, 0 < θ 1, P x 0, the olynomial defined above. Then the following estimates hold: (i) Q x 0,θ Q x 0, c(n) (θ) u P B θ (x0) x 0, (ii) Q x0, ξ c(n) D n u ξ B (x 0) where ξ R N. Proof: Using both the definitions of Q x 0,θ and of the constant c Q and Hölder s inequality we comute Q x 0,θ Q x 0, = = c Q (θ) u(x) x n Q B θ (x0) x 0, c Q (θ) u(x) xn Q B θ (x0) x 0, x n c Q (θ) 4 B θ (x0) u(x) Q x0, x n c Q (θ) B θ (x0) u(x) Q x0, x n. ] B θ (x0) x n For the second inequality we roceed analogously and aly at the end the Poincaré-inequality in the zero-boundary-data-version in order to derive: Q x 0, ξ = c Q u(x) xn ξ x n] B (x 0) c Q B (x 0) c P c Q B (x 0) u(x) ξ xn Dn u(x) ξ.

6 6 L. Beck Moreover, we will need an iteration result (cf. 18], Lemma 7.3): Lemma.5: Let ϕ: 0, ] R be a ositive non-decreasing function satisfying ϕ(θ k1 ) θ γ ϕ(θ k ) B (θ k ) n for every k N, where θ (0, 1) and γ (0, n). Then there exists a constant c deending only on n, θ and γ such that for every t (0, ] the following holds: ( t ) γϕ() ] ϕ(t) c B t γ. For the roof of the characterization of regular boundary oints we will concentrate on the model situation of a half ball and we will make use of a slight modification of Camanato s integral characterization of Hölder-continuity u to the boundary: Theorem.6 (0], Theorem.3): Consider n N, n and x 0 R n1 {0}. Suose that there are ositive constants α (0, 1], κ > 0 such that, for some v L (B 6R (x 0)), there holds the following: for all y R (x 0 ) and 4R; and { } inf v µ µ R B (y) { } inf v µ µ R B (y) ( κ ) α R ( κ ) α R for all y B R (x 0) with B (y) B R (x 0). Then there exists a Hölder-continuous reresentative v of v on B R (x 0), and for v there holds: v(x) v(z) c κ ( xz ) α R for all x, z B R (x 0), for a constant c deending only on n and α. 3 Ellitic Systems 3.1 Decay estimate The first ste in roving a regularity theorem for solutions u of ellitic systems is to establish a suitable reverse-poincaré or Cacciooli inequality. In the case of continuous coefficients a(,, ) with resect to the first two variables (instead of Hölder or Dini continuous coefficients) we have to state here the exact deendency for some linear disturbance of the weak solution u for the system div a(, u, Du) = 0 in B. (3.1) Lemma 3.1 (Cacciooli inequality): Let u W 1, (B, R N ) be a weak solution to (3.1) under the assumtions (1.), ξ R N and B (x 0 ), x 0, < 1 x 0 be an uer half ball. Then there exists a constant c = c(n, N,, L, ν) such that (1 ξ ) Du ξ e n Du ξ e n ] B / (x0) c B (x 0) (1 ξ ) u ξx n u ξx n ] c (1 ξ ) ω( ) ω( u ) ω( ξ ) ]. B (x 0) Proof: Since there holds uξ x n = 0 on, the ma η (uξ x n ) with η C0 (B (x 0 ), 0, 1]) a standard cut-off function may be taken as a test function in the weak formulation of (3.1). Now we refer to the roof of the Cacciooli inequality for the interior case, see 15], Proosition 3.1.

7 Boundary regularity for ellitic roblems 7 In the next ste we define the excess functionals analogously to 15], Section 3., in a boundary version (i. e., relacing full balls by half balls and restricting ourselves to the mean value of the normal derivative instead of the full derivative of u): For any half-ball B (x 0 ) B with x 0, a fixed function u W 1, (B, R N ) and ξ R N we define the Camanato-tye excess C(x 0, ) := the Morrey-tye excess Du (Dn u) x0, e n B (x 0) (1 (D n u) x0, ) Du (D nu) x0, e n ] (1 (D n u) x0, ), M(x 0, ) := β D n u for β (0, ) B (x 0) and finally the excess functional E(x 0, ) := C(x 0, ) ω(m(x 0, )) ω(). The next roosition rovides a suitable decay estimate, under the assumtion that the excess E(x 0, ) and the radius are sufficiently small, and will be an essential tool for the iteration later on. Proosition 3. (cf. 15], Proostion 3.): For each β (0, ) and θ (0, 1 4 ) there exist two ositive numbers ( ) ε 0 = ε 0 n, N,, ν, L, β, θ, µ( ) > 0 and ε1 (n,, β, θ) > 0 (3.) such that the following is true: If u W 1, (B, R N ) is a weak solution to (3.1) under the assumtions (1.), and if B (x 0 ), x 0, < 1 x 0, is a half ball satisfying the smallness conditions then we have for a constant c deending only on n, N,, ν and L. E(x 0, ) < ε 0 and < ε 1, (3.3) C(x 0, θ) c θ E(x 0, ) (3.4) Proof: In the first ste, we deduce an aroximate A-harmonicity result following the estimates in the roof of 15], Proosition 3., Ste 1 and obtain: for every B (x 0 ), x 0, < 1 x 0, and all functions ϕ C0(B 1 (x 0 ), R N ) with Dϕ L (B (x 0)) 1 there holds D z a ( )( x 0, 0, (D n u) x0, e n Du (Dn u) x0, e n, Dϕ ) B (x 0) c ( 1 (D n u) x0, ) 1 µ ( E(x0, ) ) 1 E(x 0, ) ] E(x 0, ) 1 E(x0, ) 1 1 ], and the constant c deends only on n, N, and L (note µ( ) 1 and the fact that u = 0 on in order to aly the Poincaré inequality in the boundary version). Now we define A := D za(x 0, 0, (D n u) x0, e n ) (1 (D n u) x0, ), w := u (D n u) x0, x n E(x0, ) (1 (D n u) x0, ), H(t) := µ( t) 1 t] 1 t 1 1 ] ; (3.5) we note here that A fulfills condition (.1), i. e., it is bounded from below and above, and further, by the definition of the excess functional E(x 0, ), there holds B (x Dw 0) 1. These definitions enable us to rewrite the revious estimate after a rescaling argument: A ( Dw, Dϕ ) c 1 (n, N,, L) H(E(x 0, )) Dϕ L B (B (x 0)) (x 0)

8 8 L. Beck for all ϕ C 1 0(B (x 0 ), R N ). For ε > 0 to be determined later, we now take δ = δ(n, N, ν, L, ε) to be the corresonding constant from the A-harmonic aroximation Lemma.. Provided that the smallness condition H(E(x 0, )) δ/c 1 (SC.1) holds, we find, according to Lemma., an A-harmonic ma h W 1, (B (x 0 ), R N ) such that B (x 0) Dh 1 and w h ε, (3.6) B (x 0) and by Lemma.3 on A-harmonic mas h is indeed smooth and satisfies, due to the last line, the estimate su B / (x0) D h c h (n, N, ν, L). We now consider θ (0, 1 4 ) fixed, to be secified later, and deduce from Taylor s theorem (kee in mind h = 0 on (x 0 )): su h(x) h(x 0 ) Dh(x 0 )(x x 0 ) = x B θ (x0) su h(x) Dn h(x 0 )x n x B θ (x0) c h (θ) 4 = c θ 4, (3.7) and the constant c deends only on n, N, ν and L. (3.6) and (3.7) now ensure that (θ) B θ (x0) w(x) D n h(x 0 )x n (θ) ( B θ (x0) w(x) h(x) (θ) ( (θ) n ε c θ 4 ) c(n, N, ν, L) θ ) h(x) D n h(x 0 )x n B θ (x0) where we have chosen ε = θ n4 in the last inequality. By the definition of w we easily conclude (θ) B θ (x0) u(x) (Dn u) x0, x n E(x 0, ) (1 (D n u) x0, ) D n h(x 0 )x n c(n, N, ν, L) θ (1 (D n u) x0, ) E(x 0, ). (3.8) Denoting by Q x 0,θ the value minimizing the functional Q B (x 0) uq x n amongst all Q R N, and P x 0, = Q x 0, x n we obtain from the last inequality that (θ) u P B θ (x0) x 0,θ c θ (1 (D n u) x0, ) E(x 0, ) (3.9) with c still deending only on n, N, ν and L. To derive a corresonding estimate with exonent instead of in (3.9) (in the case > ) we use an interolation argument. To this end we define the usual Sobolev-Exonent (i. e. = n n if < n and > arbitrary if n) and choose t (0, 1) such that 1 = 1t t 1 is satisfied. Using the inequalities of Hölder and of Sobolev-Poincaré and (3.9) we infer u P B θ (x0) x 0,θ ( u P B θ (x0) x 0,θ ) (1t) ( B θ (x0) u P c θ (t) (1 (D n u) x0, ) (1t) E(x 0, ) (1t) x 0,θ ) t ( B θ (x0) Du DP x 0,θ ) t

9 Boundary regularity for ellitic roblems 9 where c deends now on n, N,, ν and L. Alying Minkowski, Lemma.4 and the Poincaré-inequality at the boundary (kee in mind that P x 0, = Q x 0,x n vanishes on ) we obtain for the latter integral ( ) Du DP 1 B θ (x0) x 0,θ ( Du (Dn u) x0, e n B θ (x0) ( Du (D n u) x0, e n B θ (x0) c(n, ) θ n ) 1 ) 1 ( ) 1 Du (Dn u) x0, e n B (x 0) = c(n, ) θ n (1 (Dn u) x0, ) E(x 0, ) 1. Hence, inserting this in the inequality above, we get (θ) u P B θ (x0) x 0,θ c θ (1t)nt E(x 0, ) We now assume the additional smallness assumtion (Dn u) x0, Q x 0,θ ( ) 1 c(n) D n u (D n u) x0, B θ (x0) (1t) (1 (D n u) x0, ) E(x 0, ). E(x 0, ) θ (t1)tn] (1t)() if >. (SC.) One easily checks that this condition becomes void when aroaches from above, because then the exonent becomes (in deendency of the dimension n) negative. We finally arrive at (θ) u P B θ (x0) x 0,θ c θ (1 (D n u) x0, ) E(x 0, ) (3.10) with a constant c = c(n, N,, ν, L). In the end, we want to roduce mainly via Cacciooli s inequality and dividing by (1 (D n u) x0, ) the Camanato Excess quantity C(x 0, θ). Hence we have to estimate (D n u) x0,e in terms of (D n u) x0,θ in the following form: 1 (D n u) x0,e ( 1 (D n u) x0,θ ) for all θ, ]. (3.11) To this end, we comute for all θ, ] via the minimizing roerty of the meanvalue: 1 (D n u) x0,e 1 (D n u) x0,θ (D n u) x0,e (D n u) x0,θ ( ) n ( ) 1 D n u (D n u) x0,e θ B e (x0) θ n Therefore if we assume the smallness condition ( D n u (D n u) x0, B (x 0) ) 1 1 (D n u) x0,θ 1 (D n u) x0,θ θ n E(x0, ) (1 (D n u) x0, ) 1 (D n u) x0,θ. E(x 0, ) θ n (SC.3) we obtain firstly by absortion in a standard way the result (3.11) in the secial case =. Secondly we consider θ, ] arbitrary and now take into account (3.11) for = to infer (3.11) for all ossible choices of. Hence we may rewrite (3.9) and (3.10) in the following form: (θ) u P B θ (x0) x 0,θ c θ (1 (D n u) x0,θ ) E(x 0, ) (θ) u P B θ (x0) x 0,θ c θ (1 (D n u) x0,θ ) E(x 0, ) for a constant c deending only on n, N,, ν and L. (3.1)

10 10 L. Beck In the last ste we have to derive the full decay estimate for the Camanato-tye excess C(x 0, ). We aly the Cacciooli inequality (Lemma 3.1) with the choice ξ = Q θ to derive (1 Q B θ (x0) x ) 0,θ Du Q x e 0,θ n Du Q x e ] 0,θ n c (1 Q u Q B θ (x0) x 0,θ ) x x 0,θ n u Q x x 0,θ n ] θ θ c (1 Q x 0,θ ) ω( ) ω( u ) ω( Q B θ (x0) x 0,θ ) ] (3.13) with c = c(n, N,, ν, L). We will now get the exressions comrising ω and Q x 0,θ, which aear in the last formula, under control. Using Lemma.4 we have (kee in mind 1): Q x 0,θ ( Q x (D 0,θ nu) x0,θ (D n u) x0,θ ) c ( ) D n u (D n u) x0, D n u B θ (x0) B θ (x0) c θ n ( (1 (D n u) x0, ) D n u (D n u) x0, B (x 0) c (n) θ n( E(x 0, ) E(x 0, ) D n u β B (x 0) (1 (D n u) x0, ) B (x 0) D n u B (x 0) ) D n u B (x 0) ) D n u D n u B (x 0) M(x 0, ) (3.14) rovided that the smallness conditions ( E(x 0, ) θn θ n ) 1 β and c c (SC.4) hold true. We further have, by ossibly increasing the value of c, but keeing the deendency on only n, via Poincaré s inequality (Lemma.1) and (SC.4) u (θ) n u c θ n D n u M(x 0, ) ; B θ (x0) B (x 0) B (x 0) combined with the concavity of ω( ) this imlies immediately ( ) ω( u ) ω u ω(m(x 0, )) B θ (x0) B θ (x0) meaning that we have (note that ω is sublinear) B θ (x0) ω( ) ω( u ) ω( Q x 0,θ ) ] ω() ω(m(x 0, )). (3.15) Now it still remains to bound Q x 0,θ in terms of (D nu) x0,θ. Using again Lemma.4, the smallness condition (SC.4) (ossibly increasing c ), and (3.11) with = θ and =, resectively, we see that Q x 0,θ Q x 0,θ (D nu) x0,θ (D n u) x0,θ c (n) (1 (D n u) x0,θ ) θ n D n u (D n u) x0, B (x 0) (1 (D n u) x0, ) ( 1 (D n u) ) x0,θ 3 ( 1 (D n u) x0,θ ). (3.16) Hence, if we emloy the following smallness estimate ω(m(x 0, )) ω() E(x 0, ) θ n θ being derived from the latter smallness condition (SC.4) and combine this with (3.16) and (3.15), we

11 Boundary regularity for ellitic roblems 11 may estimate the second integral on the right-hand side of (3.13) by (1 Q x 0,θ ) ω( ) ω( u ) ω( Q B θ (x0) x 0,θ ) ] c θ ( 1 (D n u) x0,θ ) E(x0, ). (3.17) Next we turn to the left-hand side of (3.13) and find for > using Young s inequality and Lemma.4 ( 1 (Dn u) ) x0,θ Du (Dn u) x0,θ e n B θ (x0) c() (1 Q B θ (x0) x 0,θ ) Du (D n u) x0,θ e n Du (D n u) x0,θ e n ] c() (D n u) x0,θ Q x 0,θ c() Q x 0,θ Q x 0,θ c(n, ) (1 Q B θ (x0) x 0,θ ) Du (D n u) x0,θ e n Du (D n u) x0,θ e n ] c(n, ) (θ) u P B θ (x0) θ. (3.18) If we take into account the following two inequalities Du (D n u) x0,θ e n Du Q B θ (x0) B θ (x0) x e 0,θ n and Du (D n u) x0,θ e n Du Q B θ (x0) B θ (x0) x e 0,θ n, we further calculate combining (3.18) with (3.13) and (3.17) ( 1 (Dn u) ) x0,θ Du (Dn u) x0,θ e n Du (D n u) x0,θ e n ] B θ (x0) c B θ (x0) (1 Q x 0,θ ) u Q x 0,θ x n θ c θ ( 1 (D n u) x0,θ ) E(x0, ) u Q x x 0,θ n ] θ c θ ( 1 (D n u) x0,θ ) E(x0, ) (3.19) where we took advantage of (3.1) and (3.16) in the last inequality and where the constant c deends only on n, N,, ν and L. Dividing both sides by (1 (D n u) x0,θ ) and taking into account the definition of C(x 0, θ) this is exactly the desired excess decay estimate stated in the roosition rovided that all smallness conditions (SC.1), (SC.) and (SC.4) hold true (observe that the smallness assumtion (SC.3) is weaker than (SC.4)). The deendency of the constants ε 0 and ε 1 claimed in (3.) is now obtained by taking into consideration the deendencies in all the smallness conditions needed within the roof. 3. Proof of Theorem 1.1 In what follows we are going to combine the results concerning the decay estimates of the interior situation in 15], Proosition 3., and the boundary situation, Proosition 3., in which the constants c and smallness arameters ε 0 and ε 1 have the same deendencies (at least for the model situation). Therefore, we may assume without loss of generality that both roositions are valid for the same set of arameters. Ste 1: Choice of the constants. We fix β (0, ) and α (0, 1). We choose γ = γ(α) (n, n) such that α = 1 n γ (3.0)

12 1 L. Beck and θ (0, 1 4 ) such that θ := min {( 1 1 ( β 1, 4) ) 1 4 c ( 1 1 nγ, 4) }, (3.1) for c being the constant according to Proosition 3.. This choice fixes θ in deendency of n, N,, ν, L, α and β. We further fix a constant ε and an iteration quantity ε it 1: { θ n ε := min 4, ε 0 }, ε it := θn n (3.) where ε 0 aears in Proosition 3., and therefore they deend on n, N,, ν, L, α and β, and ε additionally on µ( ). Next, we fix δ 1 > 0 such that ω(δ1 ) < ε it ε (3.3) (note that this imlies due to the monotonicity of ω that ω(t) < ε it ε whenever t 0, δ 1 ]). This fixes δ 1 in deendency of n, N,, ν, L, α, β, µ( ) and ω( ). Lastly we define the maximally admissible radius m := min { δ 1 β 1, δ 1, ε 1 } > 0. (3.4) Here, ε 1 is the radius from Proosition 3. and m deends on n, N,, ν, L, α, β, µ( ) and ω( ). For what follows we will always assume m < 1. Ste : An almost BMO-estimate. We now consider a boundary oint x 0 and a radius min{ m, 1 x 0 } for which C(x 0, ) < ε it ε and M(x 0, ) < ε it δ 1 (3.5) is satisfied for some iteration arameter ε it ε it, 1]. Without loss of generality we may assume x 0 = 0. We shall now show that due to the choice of constants above and due to the decay estimate in Proosition 3. this smallness estimate is also valid on smaller radii, namely for every k N 0 there holds C(0, θ k ) < ε it ε and M(0, θ k ) < ε it δ 1. (I) We shall establish (I) k by induction: k = 0 is given by (3.5), and therefore we assume (I) k and rove (I) k1. We begin by noting that by definition of C(0, θ k ) and the assumtion (I) k of the induction we calculate Du (D n u) 0,θk e n ( 1 (D n u) 0,θk ) C(0, θ k ) B θ k < ε it ε ε it ε D n u. (3.6) B θ k The latter estimate enables us to derive the second inequality in (I) k1 exloiting the choices of θ, ε and m in (3.1), (3.) and (3.4), resectively: M(0, θ k1 ) (θ k1 ) β Du (D n u) 0,θ k e n (θ k1 ) β (D n u) 0,θ k B θ k1 θ βn (θ k ) β B θ k Du (D n u) 0,θ k e n θ β (θ k ) β D n u B θ k < 4 θ βn ε it ε ( (θ k ) β M(0, θ k ) ) θ β M(0, θ k ) 3 θ β M(0, θ k ) ε it θ β β ε it δ 1. Moreover, the first inequality in (I) k1 is a direct consequence of Proosition 3.: the first assumtions in (3.3) is satisfied since (I) k and the choices in (3.), (3.3) and (3.4) imly E(0, θ k ) = C(0, θ k ) ω(m(0, θ k )) ω(θ k ) ε it ε ω(δ 1 ) ω() < 3 ε it ε < ε 0,

13 Boundary regularity for ellitic roblems 13 and the second assumtions in (3.3) is fulfilled by the choice of m in (3.4). Hence, the statement of Proosition 3. imlies, taking into account (3.1), the following inequality: C(0, θ k1 ) c θ E(0, θ k ) 3 c θ ε it ε < ε it ε which finishes the roof of (I) k1 such that (I) holds for every k N 0. Ste 3: Iteration. We still consider 0 and a radius m such that (3.5) (and hence (I) for all k N 0 ) is satisfied. Then the calculation (3.6) above and the choices of ε in (3.) and of θ in (3.1) yield with ω n = B 1 that Du ωn (θ k1 ) n (Dn u) 0,θk Du (Dn u) 0,θk e n B θ k1 B θ k1 (θ n ε ) D n u ω n ε (θ k ) n 3 θ nγ θ γ B θ k D n u ω n (θ k ) n θ γ B θ k Du ω n (θ k ) n B θ k for γ defined via equation (3.0) and where we have neglected the factor ε it 1. Setting ϕ(t) := Du the last inequality may be rewritten by B t ϕ(θ k1 ) θ γ ϕ(θ k1 ) ω n (θ k ) n and the alication of the iteration Lemma.5 yields ϕ(t) c 3 ( t ) γ ϕ() t γ ] for all t for a constant c 3 = c 3 (n, N,, ν, L, α, β), i. e. there holds Du c ] 3 B γ Du 1 t γ for all t. (3.7) B t Ste 4: Hölder continuity at boundary oints. Now we are going to combine the estimates in the interior and at the boundary. For fixed α (0, 1) we define Reg,α u := { x 0 : u C 0,α (U(x 0 ) B, R N ) for some neighbourhood U(x 0 ) of x 0 }. and Sing,α u := \ Reg,α u. Now we set { ( εit ε ) } s := min ε it δ 1, (3.8) and choose a oint x 0, w.l.o.g. x 0 = 0, for which the following two estimates hold true: Du (D n u) 0, e n lim inf 0 B (0) (1 (D n u) 0, ) < s and lim inf β D n u < s. 0 B (0) The aim is now to show that 0 Reg,α u, i. e., that u is Hölder continuous with exonent α in a neighbourhood of 0 in B. For this urose we first determine a radius 0 m 6 such that Du (D n u) 0,60 e n B 6 (0) (1 (D n u) 0,60 ) < s and (6 0 ) β D n u < s. B 0 6 (0) 0 (which is ossible due to the two estimates above). Then, taking into account the definitions of C(0, 6 0 ), M(0, 6 0 ) and the arameter s < 1 in (3.8), a straightforward calculation yields C(0, 6 0 ) < ε it ε and M(0, 6 0 ) < ε it δ 1,

14 14 L. Beck and therefore, the assumtions in (3.5) of ste are satisfied for x 0 = 0 such that also C(0, 6 θ k 0 ) < ε it ε and M(0, 6 θ k 0 ) < ε it δ 1 (3.9) is fulfilled for all k N 0. In the remainder of the roof we will take advantage of the following fact which is derived analogously to (3.11) in the roof of Proosition 3.: whenever we have two (half-) balls B () (x ) B 1 (x 1 ) (with x 1 ), for which C(x 1, 1 ) θn 4 (cf. (SC.3)) and 1 θ is satisfied, then we have: 1 (D n u) x1, 1 ( 1 (D n u) x, ). (3.30) This allows us to estimate the Camanato-tye and the Morrey-tye excess in 0 also for intermediate radii: for any (0, 6 0 ] there exists a unique k N 0 such that there holds 6 θ k1 0 < 6 θ k 0. Alying (3.30) with the centre x 1 = x = 0 and radii 1 = 6 θ k 0. = we find: C(0, ) = B e B e ] Du (Dn u) 0,e e n (1 (D n u) 0,e ) Du (D nu) 0,e e n (1 (D n u) 0,e ) Du (D nu) 0,6θ k 0 e n (1 (D n u) 0,6θk 0 ) Du (D nu) 0,6θk 0 e n (1 (D n u) 0,6θk 0 ) θ n C(0, 6θ k 0 ) < θ n ε it ε (3.31) ( 6θ M(0, ) = β D n u k ) nβ 0 (6θ k 0 ) β D n u B B e 6θ k 0 ] θ βn M(0, 6θ k 0 ) < θ n ε it δ 1. (3.3) Similar to 0], , we now have to show decay estimates on a variety of balls B (y) and half balls B (y): Case 1: y 0, y 4 0 : Here, we may comare the excess functionals in B (y) via (3.30) and (3.31) with the excess functionals in B y and we obtain similar to the last comutation C(y, ) n C(0, y ) < 4n θ n ε it ε, M(y, ) nβ M(0, y ) < n θ n ε it δ 1 and via (3.7) we find B (y) Du B y Du γ c ] 3 (6 0 ) γ Du 1 γ. (3.33) B Case : y 0, 0 < < y 0 : Here we have to verify that the assumtions (3.5) for the iteration are satisfied for the half-ball B 0 (y) (kee in mind θ (0, 1 4 ) in order to aly (3.30)): C(y, 0 ) 3 n C(0, 6 0 ) < 3 n ε it ε, M(y, 0 ) 3 nβ M(0, 6 0 ) < 3 n ε it δ 1. i. e., (3.5) is valid for the iteration arameter ε it := 3 n ε it 1 by definition of ε it. Therefore we may conclude for all k N 0 there holds C(y, θ k 0 ), M(y, θ k 0 ) < 3 n ε it ε. From the calculations in (3.31) and (3.3) we now easily infer that the excess functionals for intermediate radii can only be increased by the factor θ n such that for all radii (0, 0 ] we have C(y, ) < 4 3 n θ n ε it ε and M(y, ) < 3 n θ n ε it δ 1. Moreover, as a consequence of (3.5) on the half-ball B 0 (y), we note that due to the calculations in ste 3 (see (3.7)) there holds B (y) Du c 3 ( 0 ) γ ] Du 1 B γ. (3.34)

15 Boundary regularity for ellitic roblems 15 Case 3: y B 0, B (y) B 0 : Let y = (y 1,..., y n1, 0) be the rojection of y onto R n1 {0}. Here we have the inclusions B (y) B yn (y) B y n (y ). We shall now show that the assumtions for the iteration and thus for the excess-decay estimate in the interior (see 15], (3.50) and (3.57)), which are analogous to (3.5) and (3.7), resectively, are satisfied on the ball B yn (y). If y y n ( 4 0 ) we can aly case 1 with centre y and radius y n, otherwise if y n < y < 0 we can aly case (note that we have in articular B y n (y ) B 0 (y )) and we obtain for both cases C(y, y n ) < 4 3 n θ n ε it ε and M(y, y n ) < 3 n θ n ε it δ 1. Then, recalling (3.30) and the definition of ε it, we arrive at the conclusion that Du (Du)y,yn C(y, y n ) := B yn (y) (1 (Du) y,yn ) Du (Du) y,y n ] (1 (Du) y,yn ) Du (D nu) y,y n e n B yn (y) (1 (D n u) y,y n ) Du (D nu) y,y n e n ] (1 (D n u) y,y n ) Moreover, we have M(y, y n ) := y β n B (y) n C(y, y n ) < n θ n ε it ε ε. D n u n M(y, y n ) < 6 n θ n ε it δ 1 δ 1. B yn (y) Hence, the smallness conditions for the iteration in the interior are satisfied, and ste in the Proof of 15], Theorem 1.1, combined with (3.33) and (3.34), resectively, yields ] Du c 3 yn γ Du 1 γ c 3 y γ n B yn (y) B yn (y ) ] Du 1 γ c 3 c ] 3 γ Du γ (3.35) 0 B where the constant c 3 = c 3 (n, N,, ν, L, α, β) denotes the corresonding constant aearing in the interior. Combining (3.33), (3.34) and (3.35) and alying Poincaré s inequality on the left-hand side of each inequality, we may aly Theorem.6 to conclude: u C 0,α (B 0, R N ) for α = 1 nγ, and therefore, 0 Reg,α u. This means, we have roved so far a model analogon of Theorem 1.1 (in the sequel, we will now denote by ṽ the solution of the corresonding roblem on a half-ball), i. e., we have Theorem 3.3: Let and ṽ W 1, (B, R N ) be a weak solution of system (3.1) under the assumtions (1.). Then if y Reg,α ṽ there holds: for every α (0, 1) there exists s > 0 deending only on n, N,, ν, L, α, β, ω( ) and µ( ) such that for every β (0, ) the following inclusion holds: { Sing,α ṽ y 0 : lim inf 0 or lim inf 0 Dṽ (D n ṽ) B B (y 0) e n (1 (D n ṽ) B B (y 0) ) s } D n ṽ s. B B (y 0) B B (y 0) β Ste 5: Transformation of the system to the model situation. In the next ste we sketch for convenience of the reader why the handling of the model case of a half ball is sufficient in order to deduce a criterion for a weak solution of a general ellitic system of tye (1.1) with boundary data g to be regular in the neighbourhood of a given boundary oint z Ω. Without loss of generality we may assume z = 0 and ν Ω (z) = e n where ν Ω (z) denotes the inward-ointing unit normal vector to Ω in z. The regularity assumtion on Ω ensures the existence of a function h: R n1 R of class C 1 with modulus

16 16 L. Beck of continuity τ( ), satisfying h(0) = 0 and h(0) = 0, and the existence of a radius r > 0 such that Ω B r = {x B r : x n > h(x )}. For ease of notation also the modulus of continuity of Dg is denoted by τ( ). We further choose r sufficiently small such that h(x ) < 1 for all x B r. For the functions T(x) = ( x, x n h(x ) ) introduced in Section and its inverse T 1 (y) = ( y, y n h(y ) ) (both of class C 1 with modulus of continuity τ( )) we thus obtain Lischitz constants between 1 and as well as DT, DT 1. Furthermore, we have det DT = 1 = det DT 1 and the inclusions B / T(Ω B ) B. for all r (cf. e. g. 19], Chat. 3.7). Note that this also imlies B / Ω B B. Setting ṽ(y) := u T 1 (y) g T 1 (y) allows us to calculate that ṽ W 1, (B r, R N ) is weak solution of for coefficients ã(,, ) defined by div ã(, ṽ, Dṽ) = 0 in B r ã(y, v, z) := a ( T 1 (y), v g(y), z DT(T 1 (y)) Dg(T 1 (y)) ) DT t (T 1 (y)) for all (y, v, z) B r R N R nn. Keeing in mind the the assumtions on a(,, ) given in (1.) we easily calculate that the new coefficients satisfy structure conditions analogous to (1.), namely that there holds for all y, ȳ B r, v, v R N and z, z, C R nn : ã(y, v, z) L (1 Dg ) 1 ( 1 z ) 1, Dz ã(y, v, z) L (1 Dg ) ( 1 z ), D z ã(y, v, z) (C, C) ν (1 Dg ) ( 1 z ) C, ( ã(y, v, z) ã(ȳ, v, z) ) L ( ) ( ) 1 ( ( ) ( )) 1 Dg 1 z ω y ȳ v v τ y ȳ, D z ã(y, v, z) D z ã(y, v, z) 3 L (1 Dg ) ( 1 1 z z ) ( ) µ z z. 1 z z Therefore, the corresonding assumtions in (1.) are also satisfied for the new coefficients ã(,, ) for L = L c(, Ω, g), ν = ν c(, g) and ω( ) = ω( ) τ( ). Ste 6: Transformation of the smallness conditions and final conclusion. In the last ste there still remains to show that the smallness conditions in Theorem 3.3 for the transformed system are satisfied rovided that the smallness conditions required in Theorem 1.1 are fulfilled. We still assume z = 0, ν Ω (z) = e n and use the notation introduced in ste 5. Keeing in mind D n T 1 = 1, hence D n (u T 1 )(y) = D n u(t 1 (y)), and the inclusion T 1 (B ) Ω B we find for the second of the smallness conditions β D n ṽ Ω B B B c(n) β D n (u g) Ω B ( ] ) β D n u β Dg. (3.36) Ω B Choosing s sufficiently small and using the change of variables formula we may roceed similar to (3.11) to obtain 1 (D n u) Ω B (D n ṽ) B c( Dg ) ( 1 (D n ṽ) B ) (3.37) / for all r. The latter inequality combined with the alication of Cacciooli s and Poincaré s inequality (in the boundary version) yields for the first of the smallness conditions in Theorem 3.3 that Dṽ (D n ṽ) B e n D / n ṽ (D n ṽ) B B (1 (D / n ṽ) B ) c B (1 (D / n ṽ) B ) D nṽ (D n ṽ) B ] (1 (D n ṽ) B ) c ω( ) ω( ṽ ) ω( (D n ṽ) B ) ] (3.38) B /

17 Boundary regularity for ellitic roblems 17 for a constant c deending only on n, N,, ν, L and Dg. Moreover, alying the diffeomorhism T and (3.37) we infer D n ṽ (D n ṽ) B B (1 (D n ṽ) B ) c D n ((u g) T 1 ) (D n ((u g) T 1 )) B B (1 (D n u) Ω B ) c Ω B D n u (D n u) Ω B c τ() (3.39) (1 (D n u) Ω B ) where the constant deends only on n, and Dg. Keeing in mind ω( ) ω( ṽ ) ω( (D n ṽ) B ) ] ω() n1 ω ( β B / we may combine (3.36), (3.38) and (3.39) and infer Dṽ (D n ṽ) B lim inf e n 0 B (1 (D n ṽ) B ) < s and lim inf 0 rovided that lim inf 0 β B ) D n ṽ D n ṽ < s B Du (D n u) Ω B e n Ω B (1 (D n u) Ω B ) < s and lim inf 0 β D n u < s Ω B for s sufficiently small. As a consequence of ste 4 we then obtain ṽ C 0,α locally in a neighbourhood of 0 in B, and therefore, via transformation, u C 0,α in a neighbourhood of 0 in Ω, i. e., 0 Reg Ω,α u. This finishes the roof of Theorem 1.1. Remark 3.4: Similar to the situation in the interior there are better inclusions available for the singular set in the case n. Via the Sobolev embedding we obtain that u is Hölder continuous everywhere with exonent 1 n if > n. Otherwise if = n we first deduce higher integrability of Du, i. e., Du L q1 ( Ω, R nn ) for some q 1 >, and then conclude that u is Hölder continuous everywhere with exonent 1 n q 1 (cf. 18], Remark 6.13). Hence the existence of regular boundary oints is ensured in this case, actually we have Reg Ω,α u = Ω for all α 1 n q 1. In contrast, Theorem 1.1 gives a characterization of regular boundary oints where u is locally Hölder continuous with any exonent α < 1, even though now the question of existence of regular boundary oints is oen. Moreover, we have the better inclusion for the singular set Sing Ω u given by { D ν Ω (x Sing Ω u x 0 Ω: lim inf 0)u (D ν Ω (x 0)u) Ω B(x 0) } 0 Ω B (x 0) (1 (D ν Ω (x 0)u) Ω B(x 0) ) > 0. To this aim we go back to the model situation: the alication of Cacciooli s (note ω( ) 1) and Poincaré s inequality and the Hölder continuity ṽ C 0,1 n q 1 (B, R N ) reveals β B (x 0) ( D n ṽ β B (x 0) ) D n ṽ c β ( B (x0) ṽ 1 ) c β n q 1 where the constant c deends on n, N,, L and ν. Choosing β ( n q 1, ) the left-hand side of the last inequality converges to 0 for 0, and the smallness condition on M(x 0, ) in Theorem 3.3 and hence in Theorem 1.1 is trivially satisfied. 4 Quasi-convex functionals 4.1 Decay estimate Also in the case of minimizers we consider the model case of the unit half-ball, i. e., we deal with local minimizers u W 1, (B, R N ) of the functional F B u] := F (x, u, Du). (4.1) B

18 18 L. Beck We start again by stating a Cacciooli inequality involving the exact deendency for some linear disturbance of u: Lemma 4.1 (Cacciooli inequality): Let u W 1, (B, R N ) be a local minimizer of (4.1) under the assumtions (1.4), ξ R N and B (x 0 ), x 0, < 1 x 0 be an uer half ball. Then there exists a constant c = c(n, N,, L, ν) such that (1 ξ ) Du ξ e n Du ξ e n ] B / (x0) c (1 ξ ) u ξx n u ξx n ] B (x 0) c ω( ) ω( u ) ω( ξ ) ] (1 ξ Du ). B (x 0) Proof: This lemma is roved as in the interior situation. We only have to ay attention to the alication of the quasi-convexity condition (1.4), where zero-boundary data of the testfunction ϕ is requested: hence we have to choose ϕ = η(u ξ x n ) for a standard cut-off function (for the later alication of the hole-filling argument) η C0 (B t (x 0 ), 0, 1]), 0 / s < t. The second ingredient are two higher integrability results (in order to enable an aroriate estimate for the last integral arising on the right-hand side of the Cacciooli inequality): firstly, we may rove analogously to 16], Theorem 4.1, via the alication of a Gehring-Lemma in the u-to-the-boundary version, see 11], Theorem.4: Lemma 4.: Let u W 1, (B, R N ) be a local minimizer of (4.1) under the assumtions (1.4) 1. Then there exists a higher integrability exonent q 1 > and a constant c both deending only on n, N,, ν and L such that for any half-ball B (x 0 ) B, x 0 there holds ( ) 1 Du q1 q 1 B / (x0) ( ( ) ) 1 c 1 Du. B (x 0) The second higher integrability result rovides an estimate u to the boundary (rovided that the boundary values are higher integrable), concerns solutions of functionals without (x, u)-deendency and will later be alied for the frozen functional F. For a roof we refer to the similar result 11], Lemma 3.. Lemma 4.3: Let u W 1, (B, R N ) be a local minimizer of (4.1) under the assumtions (1.4) 1, and let v 0 u W 1, 0 (B / (x 0), R N ) be a solution of the following Dirichlet roblem: v 0 min w B / (x0) G(Dw) with w u W 1, 0 (B / (x 0), R N ), (4.) where G: R nn R is continuous and satisfies ν z G(z) L(1 z ), and B / (x 0) B, x 0 a half-ball. Then there exists another higher integrability exonent q (, q 1 ] deending only on n, N,, ν and L such that ( ) 1 Dv 0 q q B / (x0) ( ) 1 ( c Dv 0 ( ) ) 1 q1 q 1 c 1 Du. B / (x0) B / (x0) In the next ste we define the excess functionals analogously to the ellitic case (cf. 15], Section 4.): For any half-ball B (x 0 ) B with x 0, a fixed function u W 1, (B, R N ) and ξ R N we define the Camanato-tye excess C(x 0, ) and as in the ellitic setting, but we redefine the Morrey-tye excess M(x 0, ) and the excess functional E(x 0, ) by setting M(x 0, ) := β Du for β (0, ) B (x 0) E(x 0, ) := C(x 0, ) ω(m(x 0, )) q q ω() q q,

19 Boundary regularity for ellitic roblems 19 where q (, q 1 ] is the higher integrability exonent introduced in Lemma 4.3. The technique for deriving a artial regularity result and the characterization of regular boundary oints now consists in comaring the minimal ma u on some half-ball B / (x 0) with the minimizer of the functional frozen in the first two variables (amongst all functions w u W 1, 0 (B / (x 0), R)). Via an aroximation Theorem based on a variational rincile due to Ekeland (see e. g. 18], Chater 5), we may now roceed as in 15], Proosition 4.4 and obtain the existence of a function v which is close with resect to the L -distance to the original minimizer u and which is an almost minimizer of the frozen functional; this rovides the following comarison result: Lemma 4.4: Let u W 1, (B, R N ) be a local minimizer of (4.1) under the assumtions (1.4), and let B (x 0 ) B, x 0, be such a half ball that E(x 0, ) 1. (4.3) Then there exists a ma v u W 1, 0 (B / (x 0), R N ) such that B / (x0) Dv Du K(x 0, ) (4.4) and ( ) 1 G(Dv) G(Dv Dϕ) c e K(x 0, ) 1 1 Dϕ, (4.5) B / (x0) B / (x0) B / (x0) for every ϕ W 1, 0 (B / (x 0), R N ), and c e is a constant deending only on n, N,, ν and L. The exonent q > is the higher integrability exonent in Lemma 4.3, and the integrand G(z) := F (x 0, 0, z) is defined by freezing F (,, z) and K(x 0, ) := ( 1 (D n u) ) ( x0, ω(m(x 0, )) q q ) ω() q q Analogously to the ellitic case we next rove the excess decay estimate for the corresonding excess quantities defined above: Proosition 4.5 (cf. 15], Proosition 4.5): For each β (0, ) and θ (0, 1 8 ) there exist two ositive numbers ( ) ε 0 = ε 0 n, N,, ν, L, β, θ, µ( ) > 0 and ε1 (n,, β, θ) > 0 (4.6) such that the following is true: If u W 1, (B, R N ) is a local minimizer to (4.1) under the assumtions (1.4), and if B (x 0 ), x 0, < 1 x 0, is a half ball satisfying the smallness conditions then we have for a constant c deending only on n, N,, ν and L. E(x 0, ) < ε 0 and < ε 1, (4.7) C(x 0, θ) c θ E(x 0, ) (4.8) Proof: In the first ste of our roof, we infer aroximate A-harmonicity (with A an adequate freezing of D zz F introduced later on). For this linearization we will use the last Proosition 4.4: Hence we assume the smallness condition (4.3), i. e., E(x 0, ) 1. (SCF-1) Then the ma v u W 1, 0 (B / (x 0), R N ) found in Proosition 4.4 minimizes the functional ξ ( ) 1 G(Dξ) c e K(x 0, ) 1 1 Dξ Dv, B / (x0) B / (x0)

20 0 L. Beck for every ξ u W 1, 0 (B / (x 0), R N ), G(z) := F (x 0, 0, z), K(x 0, ) and c e chosen according to Proosition 4.4. Deriving the Euler-Lagrange equation for this variational integral we see that there holds 0 = ( ) 1 D z G(Dv) Dϕ c e K(x 0, ) 1 1 Dϕ B / (x0) B / (x0) for all ϕ W 1, 0 (B / (x 0), R N ). Assuming ϕ C0(B 1 / (x 0), R N ) with Dϕ L (B / (x0),rn ) 1 we infer D z G(Dv) Dϕ c e K(x 0, ) 1 1, B / (x0) and therefore, introducing the abbreviation Λ := (D n u) x0, and taking into account (1.4) 4 and the fact that B / (x0) D zg(λ e n ) Dϕ = 0, we have D zz G ( ) ( Λ e n Dv Λ e n, Dϕ ) B / (x0) 1 Dzz G ( ) Λ e n Dzz G ( Λ e n t(dv Λ e n ) ) L B / (x0) 0 ( Dv Λ e n, Dϕ ) dt Dz G ( Dv ) D z G ( ) Λ e n Dϕ B / (x0) Dv Λ e n ( 1 Λ Dv Λ e n ) ( Dv Λ e n ) µ B / (x0) 1 Λ D z G ( Dv ) Dϕ B / (x0) I c e K(x 0, ) 1 1 (4.9) with the obvious labelling. To estimate I we first have to derive some reliminary estimates: via (4.4) we find Dv Λ e n c Dv Du c Du Λ e n B / (x0) B / (x0) B / (x0) c K(x 0, ) c ( 1 Λ ) C(x0, ) c ( 1 Λ ) E(x0, ) for a constant c = c(n, ), and where we used the estimate K(x 0, ) (1 Λ ) E(x 0, ) (1 Λ ) E(x 0, ) in the last line. Alying Hölder s inequality and the same calculations, we obtain the analogous result for the exonent relaced by : ( ) Dv Λ e n c Dv Du c Du Λ e n B / (x0) B / (x0) B / (x0) c(n, ) ( 1 Λ ) E(x0, ). Combining the last two estimates we find an estimate for the excess functional concerning Dv instead of Du: Dv Λ e n B / (x0) (1 Λ ) Dv Λ e n ] (1 Λ ) c v E(x 0, ), (4.10) and the constant c v deends only on the arameters n and. (4.10) now allows us to estimate I using

21 Boundary regularity for ellitic roblems 1 Hölder s and Jensen s inequality (note µ( ) 1): I = L Dv Λ e n ( 1 Λ Dv Λ e n ) µ B / (x0) c (1 Λ ) 1 Dv Λ e n ( Dv Λ e n ) µ B / (x0) (1 Λ ) 1 Λ c (1 Λ ) 1 Dv Λ e n 1 ( Dv Λ e n B / (x0) (1 Λ ) 1 µ 1 Λ c (1 Λ ) 1 ( B / (x0) Dv Λ e n (1 Λ ) ) 1 ( ( c (1 Λ ) 1 Dv Λ e n ) 1 B / (x0) (1 Λ ) c (1 Λ ) 1 ( E(x 0, ) 1 µ ( E(x0, ) ) 1 E(x 0, ) 1 B / (x0) µ c (1 Λ ) 1 E(x 0, ) 1 E(x0, ) 1 1 ] µ ( E(x0, ) ) 1, ( B / (x0) µ ( Dv Λ e n 1 Λ ) ( Dv Λ e n 1 Λ ) ( Dv Λ e n 1 Λ µ ( E(x 0, ) ) ) 1 ) ) 1 ) ) 1 where the constant c deends only on n, and L. Combining the latter estimate for I with (4.9) and we finally arrive at K(x 0, ) 1 1 (1 Λ ) 1 E(x 0, ) 1 (1 Λ ) 1 E(x 0, ) D zz G ( )( Λ e n Dv Λ e n, Dϕ ) B / (x0) c ( 1 Λ ) 1 ( µ E(x0, ) ) 1 E(x 0, ) ] E(x 0, ) 1 E(x0, ) 1 ] 1 for all ϕ C0(B 1 / (x 0), R N ) with Dϕ L (B (x0)) 1 and c = c(n, N,, ν, L). Now we define the / functions A and w analogously to the ellitic case A := D zzf (x 0, 0, Λ e n ) (1 Λ ), w := v Λ x n cv E(x 0, ) (1 Λ ) and H(t) as in (3.5); we note that A fulfills condition (.1), i. e., it is bounded from below (in the sense of Legendre-Hadamard) and above, and further, by the definition of the excess functional E(x 0, ) and the constant c v, see (4.10), there holds B / (x0) Dw 1. These definitions enable us to rewrite the revious estimate after a rescaling argument: A ( Dw, Dϕ ) c 4 (n, N,, ν, L) H(E(x 0, )) Dϕ L B / (x0) (B / (x0)) for all ϕ C0(B 1 / (x 0), R N ), which is comletely analogous to the ellitic situation, aart from the fact that we have the take the radius instead of due to the comarison technique. For ε > 0 to be determined later, we now take δ = δ(n, N, ν, L, ε) to be the corresonding constant from the A-harmonic aroximation Lemma.. Provided that the smallness condition H(E(x 0, )) δ/c 4 (SCF.) holds, we find, according to Lemma., an A-harmonic ma h W 1, (B / (x 0), R N ) such that B / (x0) Dh 1 and B / (x0) w h ε, (4.11)