A LOWER BOUND FOR THE GRADIENT OF -HARMONIC FUNCTIONS Edi Rosset. 1. Introduction. u xi u xj u xi x j
|
|
- Pearl Fowler
- 5 years ago
- Views:
Transcription
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
A Remark on -harmonic Functions on Riemannian Manifolds
Electronic Journal of ifferential Equations Vol. 1995(1995), No. 07, pp. 1-10. Published June 15, 1995. ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp (login: ftp) 147.26.103.110
More informationESTIMATES FOR SMOOTH ABSOLUTELY MINIMIZING LIPSCHITZ EXTENSIONS
Electronic Journal of Differential Equations ol. 1993(1993), No. 03, pp. 1-9. Published October 12, 1993. ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp (login: ftp) 147.26.103.110
More informationTHE HARNACK INEQUALITY FOR -HARMONIC FUNCTIONS. Peter Lindqvist and Juan J. Manfredi
Electronic Journal of ifferential Equations Vol. 1995(1995), No. 04, pp. 1-5. Published April 3, 1995. ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp (login: ftp) 147.26.103.110
More informationSome aspects of vanishing properties of solutions to nonlinear elliptic equations
RIMS Kôkyûroku, 2014, pp. 1 9 Some aspects of vanishing properties of solutions to nonlinear elliptic equations By Seppo Granlund and Niko Marola Abstract We discuss some aspects of vanishing properties
More informationAsymptotic behavior of infinity harmonic functions near an isolated singularity
Asymptotic behavior of infinity harmonic functions near an isolated singularity Ovidiu Savin, Changyou Wang, Yifeng Yu Abstract In this paper, we prove if n 2 x 0 is an isolated singularity of a nonegative
More informationAN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-harmonic FUNCTIONS. To the memory of our friend and colleague Fuensanta Andreu
AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-harmonic FUNCTIONS JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI Abstract. We characterize p-harmonic functions in terms of an asymptotic mean value
More informationSYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.
More informationAsymptotic Behavior of Infinity Harmonic Functions Near an Isolated Singularity
Savin, O., and C. Wang. (2008) Asymptotic Behavior of Infinity Harmonic Functions, International Mathematics Research Notices, Vol. 2008, Article ID rnm163, 23 pages. doi:10.1093/imrn/rnm163 Asymptotic
More informationMultiple positive solutions for a class of quasilinear elliptic boundary-value problems
Electronic Journal of Differential Equations, Vol. 20032003), No. 07, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp) Multiple positive
More informationAN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-harmonic FUNCTIONS
AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-harmonic FUNCTIONS JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI Abstract. We characterize p-harmonic functions in terms of an asymptotic mean value
More informationEverywhere differentiability of infinity harmonic functions
Everywhere differentiability of infinity harmonic functions Lawrence C. Evans and Charles K. Smart Department of Mathematics University of California, Berkeley Abstract We show that an infinity harmonic
More informationSYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 12, 1998, 47 59 SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS M. Grossi S. Kesavan F. Pacella M. Ramaswamy
More informationREGULARITY FOR INFINITY HARMONIC FUNCTIONS IN TWO DIMENSIONS
C,α REGULARITY FOR INFINITY HARMONIC FUNCTIONS IN TWO DIMENSIONS LAWRENCE C. EVANS AND OVIDIU SAVIN Abstract. We propose a new method for showing C,α regularity for solutions of the infinity Laplacian
More informationOn the infinity Laplace operator
On the infinity Laplace operator Petri Juutinen Köln, July 2008 The infinity Laplace equation Gunnar Aronsson (1960 s): variational problems of the form S(u, Ω) = ess sup H (x, u(x), Du(x)). (1) x Ω The
More informationHARDY INEQUALITIES WITH BOUNDARY TERMS. x 2 dx u 2 dx. (1.2) u 2 = u 2 dx.
Electronic Journal of Differential Equations, Vol. 003(003), No. 3, pp. 1 8. ISSN: 107-6691. UL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) HADY INEQUALITIES
More informationEXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 98, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE
More informationGrowth estimates through scaling for quasilinear partial differential equations
Growth estimates through scaling for quasilinear partial differential equations Tero Kilpeläinen, Henrik Shahgholian, and Xiao Zhong March 5, 2007 Abstract In this note we use a scaling or blow up argument
More informationCOINCIDENCE SETS IN THE OBSTACLE PROBLEM FOR THE p-harmonic OPERATOR
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 95, Number 3, November 1985 COINCIDENCE SETS IN THE OBSTACLE PROBLEM FOR THE p-harmonic OPERATOR SHIGERU SAKAGUCHI Abstract. We consider the obstacle
More informationA note on W 1,p estimates for quasilinear parabolic equations
200-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Differential Equations, Conference 08, 2002, pp 2 3. http://ejde.math.swt.edu or http://ejde.math.unt.edu
More informationTHE NEUMANN PROBLEM FOR THE -LAPLACIAN AND THE MONGE-KANTOROVICH MASS TRANSFER PROBLEM
THE NEUMANN PROBLEM FOR THE -LAPLACIAN AND THE MONGE-KANTOROVICH MASS TRANSFER PROBLEM J. GARCÍA-AZORERO, J. J. MANFREDI, I. PERAL AND J. D. ROSSI Abstract. We consider the natural Neumann boundary condition
More informationEQUAZIONI A DERIVATE PARZIALI. STEKLOV EIGENVALUES FOR THE -LAPLACIAN
EQUAZIONI A DERIVATE PARZIALI. STEKLOV EIGENVALUES FOR THE -LAPLACIAN J. GARCIA-AZORERO, J. J. MANFREDI, I. PERAL AND J. D. ROSSI Abstract. We study the Steklov eigenvalue problem for the - laplacian.
More informationEXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem
Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
More information2 A Model, Harmonic Map, Problem
ELLIPTIC SYSTEMS JOHN E. HUTCHINSON Department of Mathematics School of Mathematical Sciences, A.N.U. 1 Introduction Elliptic equations model the behaviour of scalar quantities u, such as temperature or
More informationREGULARITY AND COMPARISON PRINCIPLES FOR p-laplace EQUATIONS WITH VANISHING SOURCE TERM. Contents
REGULARITY AND COMPARISON PRINCIPLES FOR p-laplace EQUATIONS WITH VANISHING SOURCE TERM BERARDINO SCIUNZI Abstract. We prove some sharp estimates on the summability properties of the second derivatives
More informationREGULARITY FOR NON-UNIFORMLY ELLIPTIC SYSTEMS AND APPLICATION TO SOME VARIATIONAL INTEGRALS. Francesco Leonetti and Chiara Musciano
Electronic Journal of Differential Equations Vol. 1995(1995), No. 6, pp. 1-14. Published June 7, 1995. ISSN 17-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp (login: ftp) 147.6.13.11
More informationNonexistence of solutions for quasilinear elliptic equations with p-growth in the gradient
Electronic Journal of Differential Equations, Vol. 2002(2002), No. 54, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Nonexistence
More informationSELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY
Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS
More informationBoundary value problems for the infinity Laplacian. regularity and geometric results
: regularity and geometric results based on joint works with Graziano Crasta, Roma La Sapienza Calculus of Variations and Its Applications - Lisboa, December 2015 on the occasion of Luísa Mascarenhas 65th
More informationRegularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains
Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains Ilaria FRAGALÀ Filippo GAZZOLA Dipartimento di Matematica del Politecnico - Piazza L. da Vinci - 20133
More informationELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 76, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ELLIPTIC
More informationEXISTENCE OF WEAK SOLUTIONS FOR A NONUNIFORMLY ELLIPTIC NONLINEAR SYSTEM IN R N. 1. Introduction We study the nonuniformly elliptic, nonlinear system
Electronic Journal of Differential Equations, Vol. 20082008), No. 119, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu login: ftp) EXISTENCE
More informationGRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS
LE MATEMATICHE Vol. LI (1996) Fasc. II, pp. 335347 GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS CARLO SBORDONE Dedicated to Professor Francesco Guglielmino on his 7th birthday W
More informationTHE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION. Juha Kinnunen. 1 f(y) dy, B(x, r) B(x,r)
Appeared in Israel J. Math. 00 (997), 7 24 THE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION Juha Kinnunen Abstract. We prove that the Hardy Littlewood maximal operator is bounded in the Sobolev
More informationA non-local problem with integral conditions for hyperbolic equations
Electronic Journal of Differential Equations, Vol. 1999(1999), No. 45, pp. 1 6. ISSN: 17-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login:
More informationUNIQUENESS OF SELF-SIMILAR VERY SINGULAR SOLUTION FOR NON-NEWTONIAN POLYTROPIC FILTRATION EQUATIONS WITH GRADIENT ABSORPTION
Electronic Journal of Differential Equations, Vol. 2015 2015), No. 83, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu UNIQUENESS OF SELF-SIMILAR
More informationSEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT
Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC
More informationNonlinear elliptic systems with exponential nonlinearities
22-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 9, 22, pp 139 147. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationPROPERTIES OF INFINITE HARMONIC FUNCTIONS RELATIVE TO RIEMANNIAN VECTOR FIELDS
LE MATEMATICHE Vol. LXIII (2008) Fasc. II, pp. 19 37 PROPERTIES OF INFINITE HARMONIC FUNCTIONS RELATIVE TO RIEMANNIAN VECTOR FIELDS THOMAS BIESKE We employ Riemannian jets which are adapted to the Riemannian
More informationBoundary value problems for the infinity Laplacian. regularity and geometric results
: regularity and geometric results based on joint works with Graziano Crasta, Roma La Sapienza Calculus of variations, optimal transportation, and geometric measure theory: from theory to applications
More informationNON-EXTINCTION OF SOLUTIONS TO A FAST DIFFUSION SYSTEM WITH NONLOCAL SOURCES
Electronic Journal of Differential Equations, Vol. 2016 (2016, No. 45, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NON-EXTINCTION OF
More informationA Necessary and Sufficient Condition for the Continuity of Local Minima of Parabolic Variational Integrals with Linear Growth
A Necessary and Sufficient Condition for the Continuity of Local Minima of Parabolic Variational Integrals with Linear Growth E. DiBenedetto 1 U. Gianazza 2 C. Klaus 1 1 Vanderbilt University, USA 2 Università
More informationPositive eigenfunctions for the p-laplace operator revisited
Positive eigenfunctions for the p-laplace operator revisited B. Kawohl & P. Lindqvist Sept. 2006 Abstract: We give a short proof that positive eigenfunctions for the p-laplacian are necessarily associated
More informationMULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN
Electronic Journal of Differential Equations, Vol. 016 (016), No. 97, pp. 1 11. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIONS
More informationSUBELLIPTIC CORDES ESTIMATES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 000-9939XX0000-0 SUBELLIPTIC CORDES ESTIMATES Abstract. We prove Cordes type estimates for subelliptic linear partial
More informationA PROPERTY OF SOBOLEV SPACES ON COMPLETE RIEMANNIAN MANIFOLDS
Electronic Journal of Differential Equations, Vol. 2005(2005), No.??, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A PROPERTY
More informationNew Identities for Weak KAM Theory
New Identities for Weak KAM Theory Lawrence C. Evans Department of Mathematics University of California, Berkeley Abstract This paper records for the Hamiltonian H = p + W (x) some old and new identities
More informationLORENTZ ESTIMATES FOR ASYMPTOTICALLY REGULAR FULLY NONLINEAR ELLIPTIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 27 27), No. 2, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu LORENTZ ESTIMATES FOR ASYMPTOTICALLY REGULAR FULLY NONLINEAR
More informationA one-dimensional nonlinear degenerate elliptic equation
USA-Chile Workshop on Nonlinear Analysis, Electron. J. Diff. Eqns., Conf. 06, 001, pp. 89 99. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu login: ftp)
More informationEXISTENCE AND REGULARITY RESULTS FOR SOME NONLINEAR PARABOLIC EQUATIONS
EXISTECE AD REGULARITY RESULTS FOR SOME OLIEAR PARABOLIC EUATIOS Lucio BOCCARDO 1 Andrea DALL AGLIO 2 Thierry GALLOUËT3 Luigi ORSIA 1 Abstract We prove summability results for the solutions of nonlinear
More informationInequalities of Babuška-Aziz and Friedrichs-Velte for differential forms
Inequalities of Babuška-Aziz and Friedrichs-Velte for differential forms Martin Costabel Abstract. For sufficiently smooth bounded plane domains, the equivalence between the inequalities of Babuška Aziz
More informationMIXED BOUNDARY-VALUE PROBLEMS FOR QUANTUM HYDRODYNAMIC MODELS WITH SEMICONDUCTORS IN THERMAL EQUILIBRIUM
Electronic Journal of Differential Equations, Vol. 2005(2005), No. 123, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) MIXED
More informationCOMPARISON PRINCIPLES FOR CONSTRAINED SUBHARMONICS PH.D. COURSE - SPRING 2019 UNIVERSITÀ DI MILANO
COMPARISON PRINCIPLES FOR CONSTRAINED SUBHARMONICS PH.D. COURSE - SPRING 2019 UNIVERSITÀ DI MILANO KEVIN R. PAYNE 1. Introduction Constant coefficient differential inequalities and inclusions, constraint
More informationMEAN VALUE PROPERTY FOR p-harmonic FUNCTIONS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 40, Number 7, July 0, Pages 453 463 S 000-9939(0)8-X Article electronically published on November, 0 MEAN VALUE PROPERTY FOR p-harmonic FUNCTIONS
More informationSYMMETRY IN REARRANGEMENT OPTIMIZATION PROBLEMS
Electronic Journal of Differential Equations, Vol. 2009(2009), No. 149, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SYMMETRY IN REARRANGEMENT
More informationu( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3)
M ath 5 2 7 Fall 2 0 0 9 L ecture 4 ( S ep. 6, 2 0 0 9 ) Properties and Estimates of Laplace s and Poisson s Equations In our last lecture we derived the formulas for the solutions of Poisson s equation
More informationStability properties of positive solutions to partial differential equations with delay
Electronic Journal of Differential Equations, Vol. 2001(2001), No. 64, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Stability properties
More informationSome lecture notes for Math 6050E: PDEs, Fall 2016
Some lecture notes for Math 65E: PDEs, Fall 216 Tianling Jin December 1, 216 1 Variational methods We discuss an example of the use of variational methods in obtaining existence of solutions. Theorem 1.1.
More informationTheory of PDE Homework 2
Theory of PDE Homework 2 Adrienne Sands April 18, 2017 In the following exercises we assume the coefficients of the various PDE are smooth and satisfy the uniform ellipticity condition. R n is always an
More informationSome Remarks About the Density of Smooth Functions in Weighted Sobolev Spaces
Journal of Convex nalysis Volume 1 (1994), No. 2, 135 142 Some Remarks bout the Density of Smooth Functions in Weighted Sobolev Spaces Valeria Chiadò Piat Dipartimento di Matematica, Politecnico di Torino,
More informationPartial regularity for fully nonlinear PDE
Partial regularity for fully nonlinear PDE Luis Silvestre University of Chicago Joint work with Scott Armstrong and Charles Smart Outline Introduction Intro Review of fully nonlinear elliptic PDE Our result
More informationSTOKES PROBLEM WITH SEVERAL TYPES OF BOUNDARY CONDITIONS IN AN EXTERIOR DOMAIN
Electronic Journal of Differential Equations, Vol. 2013 2013, No. 196, pp. 1 28. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu STOKES PROBLEM
More informationCONTINUOUS DEPENDENCE ESTIMATES FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR DEGENERATE ELLIPTIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 20022002), No. 39, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp) CONTINUOUS DEPENDENCE
More informationLERAY LIONS DEGENERATED PROBLEM WITH GENERAL GROWTH CONDITION
2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 73 81. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
More informationRegularity of Weak Solution to Parabolic Fractional p-laplacian
Regularity of Weak Solution to Parabolic Fractional p-laplacian Lan Tang at BCAM Seminar July 18th, 2012 Table of contents 1 1. Introduction 1.1. Background 1.2. Some Classical Results for Local Case 2
More informationBrunn-Minkowski inequality for the 1-Riesz capacity and level set convexity for the 1/2-Laplacian
Brunn-Minkowski inequality for the 1-Riesz capacity and level set convexity for the 1/2-Laplacian M. Novaga, B. Ruffini Abstract We prove that that the 1-Riesz capacity satisfies a Brunn-Minkowski inequality,
More informationALEKSANDROV-TYPE ESTIMATES FOR A PARABOLIC MONGE-AMPÈRE EQUATION
Electronic Journal of Differential Equations, Vol. 2005(2005), No. 11, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ALEKSANDROV-TYPE
More informationA Sharp Minimum Principle for the Problem of Torsional Rigidity
Journal of Mathematical Analysis and Applications 33, 5765 999 Article ID jmaa99969, available online at http:wwwidealibrarycom on A Sharp Minimum Principle for the Problem of Torsional Rigidity Xi-nan
More informationMULTIPLE POSITIVE SOLUTIONS FOR KIRCHHOFF PROBLEMS WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 015 015), No. 0, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE POSITIVE SOLUTIONS
More informationOPTIMAL LIPSCHITZ EXTENSIONS AND THE INFINITY LAPLACIAN. R. F. Gariepy, Department of Mathematics, University of Kentucky
OPTIMAL LIPSCHITZ EXTENSIONS AND THE INFINITY LAPLACIAN M. G. Crandall (1), Department of Mathematics, UC Santa Barbara L. C. Evans (2), Department of Mathematics, UC Berkeley R. F. Gariepy, Department
More informationMinimal Surface equations non-solvability strongly convex functional further regularity Consider minimal surface equation.
Lecture 7 Minimal Surface equations non-solvability strongly convex functional further regularity Consider minimal surface equation div + u = ϕ on ) = 0 in The solution is a critical point or the minimizer
More informationMEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW
MEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW GREGORY DRUGAN AND XUAN HIEN NGUYEN Abstract. We present two initial graphs over the entire R n, n 2 for which the mean curvature flow
More informationRegularity estimates for fully non linear elliptic equations which are asymptotically convex
Regularity estimates for fully non linear elliptic equations which are asymptotically convex Luis Silvestre and Eduardo V. Teixeira Abstract In this paper we deliver improved C 1,α regularity estimates
More informationOn non negative solutions of some quasilinear elliptic inequalities
On non negative solutions of some quasilinear elliptic inequalities Lorenzo D Ambrosio and Enzo Mitidieri September 28 2006 Abstract Let f : R R be a continuous function. We prove that under some additional
More informationTHREE SINGULAR VARIATIONAL PROBLEMS. Lawrence C. Evans Department of Mathematics University of California Berkeley, CA 94720
THREE SINGLAR VARIATIONAL PROBLEMS By Lawrence C. Evans Department of Mathematics niversity of California Berkeley, CA 9470 Some of the means I use are trivial and some are quadrivial. J. Joyce Abstract.
More informationWELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL FLUID EQUATIONS WITH DEGENERACY ON THE BOUNDARY
Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 13, pp. 1 15. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu WELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL
More informationA Dirichlet problem in the strip
Electronic Journal of Differential Equations, Vol. 1996(1996), No. 10, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp (login: ftp) 147.26.103.110 or 129.120.3.113
More informationEXISTENCE OF SOLUTIONS FOR CROSS CRITICAL EXPONENTIAL N-LAPLACIAN SYSTEMS
Electronic Journal of Differential Equations, Vol. 2014 (2014), o. 28, pp. 1 10. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTECE OF SOLUTIOS
More informationEQUIVALENCE OF VISCOSITY AND WEAK SOLUTIONS FOR THE p(x)-laplacian
EQUIVALENCE OF VISCOSITY AND WEAK SOLUTIONS FOR THE p(x-laplacian PETRI JUUTINEN, TEEMU LUKKARI, AND MIKKO PARVIAINEN Abstract. We consider different notions of solutions to the p(x-laplace equation div(
More informationNONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS INVOLVING EXPONENTIAL CRITICAL GROWTH IN R Introduction
Electronic Journal of Differential Equations, Vol. 014 (014), No. 59, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONLINEAR SCHRÖDINGER
More informationSYMMETRY AND REGULARITY OF AN OPTIMIZATION PROBLEM RELATED TO A NONLINEAR BVP
lectronic ournal of Differential quations, Vol. 2013 2013, No. 108, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SYMMTRY AND RGULARITY
More informationEXISTENCE OF STRONG SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS
EXISTENCE OF STRONG SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS Adriana Buică Department of Applied Mathematics Babeş-Bolyai University of Cluj-Napoca, 1 Kogalniceanu str., RO-3400 Romania abuica@math.ubbcluj.ro
More informationOPTIMAL CONTROL AND STRANGE TERM FOR A STOKES PROBLEM IN PERFORATED DOMAINS
PORTUGALIAE MATHEMATICA Vol. 59 Fasc. 2 2002 Nova Série OPTIMAL CONTROL AND STRANGE TERM FOR A STOKES PROBLEM IN PERFORATED DOMAINS J. Saint Jean Paulin and H. Zoubairi Abstract: We study a problem of
More informationEXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY
Electronic Journal of Differential Equations, Vol. 216 216), No. 329, pp. 1 22. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN
More informationThe Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:
Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply
More informationMath The Laplacian. 1 Green s Identities, Fundamental Solution
Math. 209 The Laplacian Green s Identities, Fundamental Solution Let be a bounded open set in R n, n 2, with smooth boundary. The fact that the boundary is smooth means that at each point x the external
More informationRELATIONSHIP BETWEEN SOLUTIONS TO A QUASILINEAR ELLIPTIC EQUATION IN ORLICZ SPACES
Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 265, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu RELATIONSHIP
More informationHOPF S LEMMA FOR A CLASS OF SINGULAR/DEGENERATE PDE-S
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 40, 2015, 475 484 HOPF S LEMMA FOR A CLASS OF SINGULAR/EGENERATE PE-S Hayk Mikayelyan and Henrik Shahgholian Xi an Jiaotong-Liverpool University,
More informationBesov regularity of solutions of the p-laplace equation
Besov regularity of solutions of the p-laplace equation Benjamin Scharf Technische Universität München, Department of Mathematics, Applied Numerical Analysis benjamin.scharf@ma.tum.de joint work with Lars
More informationOptimal Transportation. Nonlinear Partial Differential Equations
Optimal Transportation and Nonlinear Partial Differential Equations Neil S. Trudinger Centre of Mathematics and its Applications Australian National University 26th Brazilian Mathematical Colloquium 2007
More informationON THE NATURAL GENERALIZATION OF THE NATURAL CONDITIONS OF LADYZHENSKAYA AND URAL TSEVA
PATIAL DIFFEENTIAL EQUATIONS BANACH CENTE PUBLICATIONS, VOLUME 27 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WASZAWA 1992 ON THE NATUAL GENEALIZATION OF THE NATUAL CONDITIONS OF LADYZHENSKAYA
More informationPseudo-monotonicity and degenerate elliptic operators of second order
2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 9 24. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationCarleman estimates for the Euler Bernoulli plate operator
Electronic Journal of Differential Equations, Vol. 000(000), No. 53, pp. 1 13. ISSN: 107-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login:
More informationarxiv: v1 [math.fa] 26 Jan 2017
WEAK APPROXIMATION BY BOUNDED SOBOLEV MAPS WITH VALUES INTO COMPLETE MANIFOLDS PIERRE BOUSQUET, AUGUSTO C. PONCE, AND JEAN VAN SCHAFTINGEN arxiv:1701.07627v1 [math.fa] 26 Jan 2017 Abstract. We have recently
More informationBLOW-UP AND EXTINCTION OF SOLUTIONS TO A FAST DIFFUSION EQUATION WITH HOMOGENEOUS NEUMANN BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 016 (016), No. 36, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu BLOW-UP AND EXTINCTION OF SOLUTIONS TO A FAST
More informationNonexistence of solutions to systems of higher-order semilinear inequalities in cone-like domains
Electronic Journal of Differential Equations, Vol. 22(22, No. 97, pp. 1 19. ISSN: 172-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp Nonexistence of solutions
More informationREGULARITY OF GENERALIZED NAVEIR-STOKES EQUATIONS IN TERMS OF DIRECTION OF THE VELOCITY
Electronic Journal of Differential Equations, Vol. 00(00), No. 05, pp. 5. ISSN: 07-669. UR: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu REGUARITY OF GENERAIZED NAVEIR-STOKES
More informationSIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP AND UNIFORM BLOW-UP PROFILES FOR REACTION-DIFFUSION SYSTEM
Electronic Journal of Differential Euations, Vol. 22 (22), No. 26, pp. 9. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SIMULTANEOUS AND NON-SIMULTANEOUS
More informationEXISTENCE OF SOLUTIONS TO P-LAPLACE EQUATIONS WITH LOGARITHMIC NONLINEARITY
Electronic Journal of Differential Equations, Vol. 2009(2009), No. 87, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF SOLUTIONS
More informationHomogenization and error estimates of free boundary velocities in periodic media
Homogenization and error estimates of free boundary velocities in periodic media Inwon C. Kim October 7, 2011 Abstract In this note I describe a recent result ([14]-[15]) on homogenization and error estimates
More informationNOTE ON A REMARKABLE SUPERPOSITION FOR A NONLINEAR EQUATION. 1. Introduction. ρ(y)dy, ρ 0, x y
NOTE ON A REMARKABLE SUPERPOSITION FOR A NONLINEAR EQUATION PETER LINDQVIST AND JUAN J. MANFREDI Abstract. We give a simple proof of - and extend - a superposition principle for the equation div( u p 2
More information