A LOWER BOUND FOR THE GRADIENT OF -HARMONIC FUNCTIONS Edi Rosset. 1. Introduction. u xi u xj u xi x j

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1 Electronic Journal of Differential Equations, Vol. 1996(1996) No. 0, pp ISSN URL: ( ) telnet (login: ejde), ftp, and gopher access: ejde.math.swt.edu or ejde.math.unt.edu A LOWER BOUND FOR THE GRADIENT OF -HARMONIC FUNCTIONS Edi Rosset Abstract We establish a lower bound for the gradient of the solution to -Laplace equation in a strongly star-shaped annulus with capacity type boundary conditions. The proof involves properties of the radial derivative of the solution, so that starshapedness of level sets easily follows. 1. Introduction In this paper we deal with solutions to the -Laplace equation u = n u xi u xj u xi x j =0, (1 ) i,j=1 in a domain of R n. Equation (1 ) was first considered by G. Aronsson ([Ar1], [Ar]) and naturally arises as the Euler equation of minimal Lipschitz extensions. It is a highly degenerate elliptic equation which is formally the limit, as p, ofthep-laplace equation p u =div( Du p Du) =0. (1 p ) This limit process has been recently made rigorous by R. Jensen in [J], where he establishes the fundamental result that any Dirichlet problem for equation (1 )has a unique viscosity solution u W 1, () C 0 ( ) which is the limit, as p, of the unique solution u p W 1,p () to equation (1 p ) satisfying the same Dirichlet data (which exists and is unique by standard variational arguments), in the sense that u p u uniformly in and weakly in W 1,q () for any q such that q<.for a discussion of the related concepts of absolutely minimizing Lipschitz extension, variational solution and viscosity solution to equation (1 ), we also refer to [B-D- M]. Concerning the critical points of -harmonic functions, Aronsson proved that any non-constant C solution to (1 ) in the plane has non-vanishing gradient ([Ar]), this result has been recently extended to C 4 solutions in higher dimensions by L. C. Evans ([E]). On the other hand, Aronsson gave examples of C 1 non-constant (viscosity) solutions to (1 ) having an interior critical point ([Ar3]) Subject Classification: 35J70, 35B05, 35B50, 31C45. Key words and phrases: -harmonic functions, p-harmonic functions. c 1996 Southwest Texas State University and University of North Texas. Submitted: November 0, Published February 6, Partially supported by MURST.

2 Edi Rosset EJDE 1996/0 We consider a strongly star-shaped annulus, that is = \ 1, where 1 and, 1, are bounded domains with C -smooth boundaries 1 and satisfying the following strong starshapedness condition with respect to the origin α 0 > 0 such that n(x) x α 0 x, x i,i=1,, (S) where n(x) is the outer unit normal to i in x i, i=1,. We establish a lower bound for the gradient of the solution u to equation (1 ) with capacity type boundary conditions { u =0 on 1 u=1 on (BC) Let us recall that solutions to equation (1 p ) satisfying boundary conditions (BC) were studied by J. A. Pfaltzgraff in [P], proving starshapedness of the level sets, and also by J. L. Lewis in [L], who proved, having convexity results as objective, a lower bound for the gradient for any p, 1<p<. In this note we show that such a lower bound is uniform with respect to p in a neighbourhood of + and furthermore we prove that it extends to the solution to equation (1 ) satisfying boundary conditions (BC). We note that in the two dimensional case one can prove a lower bound for Du p in all of uniformly as p, without any starshapedness assumption, since log Du p satisfies a uniformly elliptic equation, due to a result of G. Alessandrini ([Al]), but unfortunately the type of convergence we have for the u p s, as described previously, does not ensure the convergence of inf Du p as p. On the other hand the starshapedness assumption becomes necessary when n 3, in fact by an example of A. W. J. Stoddart ([S]) with p =thereexist simply connected domains 1, for which the solution to problem (1 p )-(BC) has the gradient vanishing at at least one interior point. By standard arguments it can be seen that the example by Stoddart can also be generalized to the case p for any p (1, ). Notation. We shall denote the Euclidean scalar product of two elements u, v R n by u v.. Statement and proof of main result Let us briefly sketch the steps of the proof. Let u, u p denote the solutions to the Dirichlet problems (1 )-(BC) and (1 p )-(BC) respectively. We establish a lower bound for Du p on, uniformly in p, forlargep(see Lemma ). By the starshapedness assumption, this implies a lower bound on for the radial derivative v p = Du p x, uniformly in p. InLemma1weprovethatv p satisfies in the distributional sense a degenerate elliptic equation, see also the Remark following Lemma 1 and, constructing an appropriate test function, we deduce a lower bound for v p in all of. Passing to the limit as p, a lower bound for the radial derivative v = Du x follows. Finally, starshapedness of level sets is a consequence of the positivity of the radial derivative.

3 EJDE 1996/0 A Lower Bound for the Gradient 3 Lemma 1. Let u p be a weak solution to (1 p ) in a domain, where p>.let n v p =Du p x = x i u p,xi, i=1 A p =(A p ij ) i,j=1,...,n, (A p ij )= Du p p δ ij +(p ) Du p p 4 u p,xi u p,xj. Then v p satisfies the following identity A p Dv p Dφ =0, φ C0 (). () Proof. For simplicity of notations we shall drop the index p from A p, u p, v p. Let us recall that, by a well known result of K. Uhlenbeck ([U]), weak solutions to (1 p )belongtoc 1,α p loc () and satisfy Du u xi x j L loc (), so that ADv L loc (). Let φ C 0 (). Then, by equation (1 p), ADv Dφ = Du p x i Du xi Dφ + Du p Du Dφ+ + (p ) Du p 4 x i u xj u xi x j Du Dφ + (p ) Du p 4 Du Du Dφ = ( = x i Du p Du ) ( ) Dφ = Du p Du (x x i Dφ) xi i i = Du p Du Dh =0, where h = n i=1 x iφ xi +(n 1)φ. Remark. Let us point out that () means that v p is a distributional solution to the following degenerate elliptic equation div( Du p p Dv p +(p ) Du p p 4 Du p Dv p Du p )=0. The degeneracy of this equation prevents to replace C 0 with W 1, 0 test functions and therefore a maximum principle is not straightforward. The suitable maximum principle will be derived in the proof of Theorem 1, under hypotheses which ensure that a Hopf-type Lemma holds. Lemma (A uniform Hopf-type Lemma). Let = \ 1, where 1 and, 1 are two simply connected bounded domains in R n with C boundaries 1,.Letu p be the solution to (1 p )-(BC). Then there exist a constant C >0, Cindependent of p, and p>nsuch that Du p (x) C x, p p. (3)

4 4 Edi Rosset EJDE 1996/0 Proof. There exists R>0 such that for any x 0 there exists a ball of center y and radius R such that B R (y) andx 0 B R (y). Let, for instance, ) x 0 1 and y as above. Let us define, for p>n,w(x)=k (R p n p n p 1 r p 1, where r = x y, R <r<rand K is a positive constant to be chosen later. Let A = {x R < x y <R}, d ={x d(x, ) >d},ford>0. Let us consider the unique viscosity solution u to the Dirichlet problem (1 )-(BC). Then the comparison principle ([J]) and Harnack inequality ([L-M]) or viscosity solutions to equation (1 ) imply that 0 <u<1in. Letγ=max R \ 3 For Ru>0. ( ) sufficiently large p, u p γ in B R so that, choosing K = γ ( R p n p 1 R ) 1 p n p 1, we have w u p on B R. Moreover, w 0 u p on B R and p w =0inA. The comparison principle for solutions to equation (1 p ) implies that w u p in A. Since w(x 0 )=u p (x 0 )=0,wehave u p n (x 0) w r (x 0 )= γ ( ) ( p n 1 p n R 1 n p 1 R p n R p 1) p 1 γ p 1 R, as p, where n is the outer unit normal to 1, and (3) follows. Theorem 1. Let be as in the hypotheses of Lemma and moreover let us assume that the starshapedness condition (S) holds. Let u be the solution to the Dirichlet problem (1 )-(BC) extended by 0 to all of. Then there exist a constant δ>0, δindependent of p, and p>nsuch that u p (x) δ x, p p, (4) r u (x) δ a.e. x, (5) r where u p is the solution to (1 p )-(BC). Moreover the set k = {x u(x) <k} is star-shaped w. r. t. the origin, for any k (0, 1). Proof. Let us consider the unique solution u p W 1,p () to the Dirichlet problem (1 p )-(BC) and notice that u p solves a uniformly elliptic quasilinear equation in a neighbourhood of in, due to the lower bound for Du p. Therefore, by standard estimates (see [L-U]), Du p (p )/ u p,xi x j L () so that ADv p L () and () extends to test functions in W 1, 0 (). From (3) and (S) it follows that v p C := Cα 0 r 1 on, p p, (6) where r 1 = d(0, 1 ). Using the test function φ =( v p p v p C p ) W 1, 0 () in (), we get (p ) Du p p 4 v p p (Dvp Du p ) + Du p p v p p Dvp =0. (7) v p <C v p <C Let W p = {x v p <C}=U p V p, where U p = {x v p <C, v p 0}, V p ={x v p <C, v p =0}. Let U p j, j J N, be the open connected

5 EJDE 1996/0 A Lower Bound for the Gradient 5 components of U p.sincedu p 0inU p,wehaveu p C (U p ), and therefore from (7) it follows that v p c p j in U p j.from Up j {x v p =C} {x v p =0} and the definition of U p it follows that U p =. Therefore v p 0inW p.sinceis connected and (6) holds it follows that W p =, thatisv p Cin for p p. Now let C <C. If there exists a set S of positive measure such that v := Du x C in S, sincev p vweakly in L q for any q<, we get the contradiction Cµ(S) χ S v p χ S v C µ(s). Therefore v > C almost everywhere, for any C <C. We have v C almost everywhere and (4) and (5) follow with δ = C r, where r =sup x x. In order to prove that k is star-shaped w. r. t. the origin, let us assume by contradiction that there exist points x k, λx k,withλ (0, 1) and let β = u(λx) u(x) > 0. There exists p p such that u p (λx) u p (x) β > 0, but this contradicts (4). References [Al] G. Alessandrini, Isoperimetric inequalities for the length of level lines of solutions of quasilinear capacity problems in the plane, J. Appl. Math. Phys. (ZAMP) 40, (1989), [Ar1] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6, (1967), [Ar] G. Aronsson, On the partial differential equation u x u xx+u x u y u xy +u y u yy =0, Ark. Mat. 7, (1968), [Ar3] G. Aronsson, On certain singular solutions of the partial differential equation u x u xx +u x u y u xy + u y u yy =0,Manuscripta Math. 47, (1984), [B-D-M] T. Bhattacharya, E. DiBenedetto, J. Manfredi, Limits as p of p u p = f and related extremal problems, Rend. Sem. Mat. Univ. Pol. Torino, Fascicolo Speciale Nonlinear PDE s, (1989), [E] L. C. Evans, Estimates for smooth absolutely minimizing Lipschitz extensions, Electron. J. Differential Equations 1993, (1993), No. 3, 1 9. [J] R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal. 13, (1993), [L] J. L. Lewis, Capacitary functions in convex rings, Arch. Rational Mech. Anal. 66, (1977), [L-M] P. Lindqvist, J. J. Manfredi, The Harnack inequality for -harmonic functions, Electron. J. Differential Equations 1995, (1995), No. 4, 1 5. [L-U] O. A. Ladyzhenskaya, N. N. Ural tseva, Linear and quasilinear elliptic equations, Academic Press, New York, [P] J. A. Pfaltzgraff, Radial symmetrization and capacities in space, Duke Math. J. 34, (1967), [S] A. W. J. Stoddart, The shape of level surfaces of harmonic functions in three dimensions, Michigan Math. J. 11, (1964), 5 9.

6 6 Edi Rosset EJDE 1996/0 [U] K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems, Acta Math. 138, (1977), Edi Rosset Dipartimento di Scienze Matematiche Università ditrieste Piazzale Europa Trieste, Italy rossedi@univ.trieste.it

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