## The pseudo-$$p$$-Laplace eigenvalue problem and viscosity solutions as $$p\to\infty$$.(English)Zbl 1092.35074

Several results concerning the pseudo-$$p$$-Laplacian operator $\widetilde{\Delta}_p: u\mapsto\sum_{i=1}^n{\frac{\partial{}}{\partial{x_i}} \left(\left| \frac{\partial{u}}{\partial{x_i}}\right| ^{p-2}\frac{\partial{u}}{\partial{x_i}}\right)}\qquad\text{in}\quad\Omega\subset\mathbb{R}^n,\;p>1,$ analogous to those known for the $$p$$-Laplacian, are proved. The authors discuss first properties of the first eigenfunction of the eigenvalue problem \left\{ \begin{alignedat}{2} &{-\widetilde{\Delta}_p{u}}=\lambda| u| ^{p-2}u&&\qquad\text{in }\Omega,\\ &u=0&&\qquad\text{on }\partial{\Omega}, \end{alignedat} \right. namely its positivity, boundedness, uniqueness and Hölder continuity. It is shown that for $$p\geq2$$ every weak solution of the eigenvalue problem is a viscosity solution, and that any nonnegative viscosity solution is locally Lipschitz continuous.
It is derived that for $$\Omega$$ a bounded domain, the limit equation for $$p\to\infty$$ is $\min{\left\{\max_{i=1,\dots,n}{\left| \frac{\partial{u}}{\partial{x_i}}\right| -\widetilde{\Lambda}_\infty u,-\widetilde{\Delta}_\infty u}\right\}}=0,$ where $$\widetilde{\Lambda}_\infty=\lim_{p\to\infty}{\widetilde{\lambda}_p^{1/p}}$$, $$\widetilde{\lambda}_p$$ is the first eigenvalue of the eigenvalue problem, and $\widetilde{\Delta}_\infty u=\sum_{i\in I(\nabla u(x))}{\left| \frac{\partial{u}}{\partial{x_i}}\right| ^2 \frac{\partial^2{u}}{\partial{x_i^2}}},$
$I(\xi)=\left\{i\in\mathbb{N}:1\leq i\leq n,\;\max_{j=1,\dots,n}{| \xi_j| }=| \xi_i| \right\},\quad\xi\in\mathbb{R}^n,$ in the sense that every cluster point of any sequence of the first eigenfunctions corresponding to $$p_n$$, $$p_n\to\infty$$, is a viscosity solution of the limit equation. Moreover, $$1/\widetilde{\Lambda}_\infty$$ is the radius of the largest $$l^1$$ ball which fits in $$\Omega$$.
The authors then introduce some examples which show that the function d$$(x,\partial{\Omega})$$ (the $$l^1$$ distance to the boundary), which is a minimizer of the corresponding Rayleigh quotient, need not be a viscosity solution of the limit equation. A connection between the viscosity solution of \left\{ \begin{alignedat}{2} &{-\widetilde{\Delta}_\infty{u}}=0&&\qquad\text{in }\Omega,\\ &u=g&&\qquad\text{on }\partial{\Omega}, \end{alignedat} \right. and a so-called absolutely minimizing Lipschitz extension of $$g$$ into $$\Omega$$ is discussed, where in the definition of the Lipschitz continuity the $$l^1$$ metric is used instead of the Euclidean one, which is used in the case of the $$p$$-Laplacian. Finally, several concavity and symmetry results are given.

### MSC:

 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 49R50 Variational methods for eigenvalues of operators (MSC2000) 35B65 Smoothness and regularity of solutions to PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 35P15 Estimates of eigenvalues in context of PDEs
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### References:

 [1] W. Allegretto and Yin Xi Huang, A Picone’s identity for the $$p$$-Laplacian and applications. Nonlin. Anal. TMA 32 (1998) 819-830. Zbl 0930.35053 · Zbl 0930.35053 [2] A. Alvino , V. Ferone , G. Trombetti and P.L. Lions , Convex symmetrization and applications . Ann. Inst. H. Poincaré Anal. Non Linéaire 14 ( 1997 ) 275 - 293 . Numdam | MR 1441395 | Zbl 0877.35040 · Zbl 0877.35040 [3] A. Anane , Simplicité et isolation de la première valeur propre du $$p$$-laplacien avec poids . C. R. Acad. Sci. Paris Sér. I Math. 305 ( 1987 ) 725 - 728 . MR 920052 | Zbl 0633.35061 · Zbl 0633.35061 [4] A. Anane , A. Benazzi and O. Chakrone , Sur le spectre d’un opérateur quasilininéaire elliptique “dégénéré” . Proyecciones 19 ( 2000 ) 227 - 248 . [5] G. Aronsson , Extension of functions satisfying Lipschitz conditions . Ark. Math. 6 ( 1967 ) 551 - 561 . MR 217665 | Zbl 0158.05001 · Zbl 0158.05001 [6] G. Aronsson , On the partial differential equation $$u_x^2u_{xx}+2u_xu_yu_{xy}+u_y^2u_{yy}=0$$ . Ark. Math. 7 ( 1968 ) 395 - 425 . MR 237962 | Zbl 0162.42201 · Zbl 0162.42201 [7] G. Barles , Remarks on uniqueness results of the first eigenvalue of the $$p$$-Laplacian . Ann. Fac. Sci. Toulouse 9 ( 1988 ) 65 - 75 . Numdam | MR 971814 | Zbl 0621.35068 · Zbl 0621.35068 [8] G. Barles and J. Busca , Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term . Comm. Partial Differential Equations 26 ( 2001 ) 2323 - 2337 . MR 1876420 | Zbl 0997.35023 · Zbl 0997.35023 [9] M. Belloni and B. Kawohl , A direct uniqueness proof for equations involving the $$p$$-Laplace operator . Manuscripta Math. 109 ( 2002 ) 229 - 231 . MR 1935031 | Zbl 1100.35032 · Zbl 1100.35032 [10] T. Bhattacharya , E. DiBenedetto and J. Manfredi , Limits as $$p\rightarrow \infty$$ of $$\Delta _pu_p=f$$ and related extremal problems . Rend. Sem. Mat., Fasciolo Speciale Nonlinear PDE’s. Univ. Torino ( 1989 ) 15 - 68 . [11] T. Bhattacharya , An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions . Electron. J. Differential Equations 2001 ( 2001 ) 1 - 8 . MR 1836812 | Zbl 0966.35052 · Zbl 0966.35052 [12] H. Brezis and L. Oswald , Remarks on sublinear problems . Nonlinear Anal. 10 ( 1986 ) 55 - 64 . MR 820658 | Zbl 0593.35045 · Zbl 0593.35045 [13] M.G. Crandall , L.C. Evans and R.F. Gariepy , Optimal Lipschitz extensions and the infinity Laplacian . Calc. Var. Partial Differential Equations 13 ( 2001 ) 123 - 139 . MR 1861094 | Zbl 0996.49019 · Zbl 0996.49019 [14] M.G. Crandall , H. Ishii and P.L. Lions , User’s guide to viscosity solutions of second order partial differential equations . Bull. Amer. Math. Soc. (N.S.) 27 ( 1992 ) 1 - 67 . arXiv | Zbl 0755.35015 · Zbl 0755.35015 [15] Y.G. Chen , Y. Giga and S. Goto , Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations . J. Differ. Geom. 33 ( 1991 ) 749 - 786 . Article | MR 1100211 | Zbl 0696.35087 · Zbl 0696.35087 [16] J.I. Diaz and J.E. Saá , Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires . C. R. Acad. Sci. Paris Sér. I Math. 305 ( 1987 ) 521 - 524 . MR 916325 | Zbl 0656.35039 · Zbl 0656.35039 [17] E. DiBenedetto , $$C^{1+\alpha }$$ local regularity of weak solutions of degenerate elliptic equations . Nonlinear Anal. TMA 7 ( 1983 ) 827 - 850 . MR 709038 | Zbl 0539.35027 · Zbl 0539.35027 [18] A. Elbert , A half-linear second order differential equation . Qualitative theory of differential equations, (Szeged 1979). Colloq. Math. Soc. János Bolyai 30 ( 1981 ) 153 - 180 . MR 680591 | Zbl 0511.34006 · Zbl 0511.34006 [19] N. Fukagai , M. Ito and K. Narukawa , Limit as $$p\rightarrow \infty$$ of $$p$$-Laplace eigenvalue problems and $$L^\infty$$ inequality of the Poincaré type . Differ. Integral Equations 12 ( 1999 ) 183 - 206 . MR 1672746 | Zbl 1064.35512 · Zbl 1064.35512 [20] M. Giaquinta and E. Giusti , On the regularity of the minima of variational integrals . Acta Math. 148 ( 1982 ) 31 - 46 . MR 666107 | Zbl 0494.49031 · Zbl 0494.49031 [21] D. Gilbarg and N. Trudinger , Elliptic Partial Differential Equations of second Order . Springer Verlag, Berlin-Heidelberg-New York ( 1977 ). MR 473443 | Zbl 0361.35003 · Zbl 0361.35003 [22] T. Ishibashi and S. Koike , On fully nonlinear pdes derived from variational problems of $$L^p$$-norms . SIAM J. Math. Anal. 33 ( 2001 ) 545 - 569 . MR 1871409 | Zbl 1030.35088 · Zbl 1030.35088 [23] U. Janfalk , Behaviour in the limit, as $$p\rightarrow \infty$$, of minimizers of functionals involving $$p$$-Dirichlet integrals . SIAM J. Math. Anal. 27 ( 1996 ) 341 - 360 . MR 1377478 | Zbl 0853.35028 · Zbl 0853.35028 [24] R. Jensen , Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient . Arch. Rational Mech. Anal. 123 ( 1993 ) 51 - 74 . MR 1218686 | Zbl 0789.35008 · Zbl 0789.35008 [25] P. Juutinen , Personal Communications . [26] P. Juutinen , P. Lindqvist and J. Manfredi , The $$\infty$$-eigenvalue problem . Arch. Rational Mech. Anal. 148 ( 1999 ) 89 - 105 . MR 1716563 | Zbl 0947.35104 · Zbl 0947.35104 [27] B. Kawohl , Rearrangements and convexity of level sets in PDE . Springer, Lecture Notes in Math. 1150 ( 1985 ). MR 810619 | Zbl 0593.35002 · Zbl 0593.35002 [28] B. Kawohl , A family of torsional creep problems . J. Reine Angew. Math. 410 ( 1990 ) 1 - 22 . Article | MR 1068797 | Zbl 0701.35015 · Zbl 0701.35015 [29] B. Kawohl , Symmetry results for functions yielding best constants in Sobolev-type inequalities . Discrete Contin. Dynam. Systems 6 ( 2000 ) 683 - 690 . MR 1757396 | Zbl pre01492653 · Zbl 1157.35342 [30] B. Kawohl and N. Kutev , Viscosity solutions for degenerate and nonmonotone elliptic equations , edited by B. da Vega, A. Sequeira and J. Videman. Plenum Press, New York & London, Appl. Nonlinear Anal. ( 1999 ) 185 - 210 . MR 1727452 | Zbl 0960.35040 · Zbl 0960.35040 [31] O.A. Ladyzhenskaya and N.N. Ural’tseva , Linear and quasilinear equations of elliptic type ,Second edition, revised. Izdat. “Nauka” Moscow ( 1973 ). English translation by Academic Press. [32] G.M. Lieberman , Gradient estimates for a new class of degenerate elliptic and parabolic equations . Ann. Scuola Normale Superiore Pisa Ser. IV 21 ( 1994 ) 497 - 522 . Numdam | MR 1318770 | Zbl 0839.35018 · Zbl 0839.35018 [33] P. Lindqvist , A nonlinear eigenvalue problem . Rocky Mountain J. 23 ( 1993 ) 281 - 288 . Article | MR 1212743 | Zbl 0785.34050 · Zbl 0785.34050 [34] P. Lindqvist , On the equation div$$(|\nabla u|^{p-2}\nabla u)+ \Lambda |u|^{p-2}u$$ $$=0$$ . Proc. Amer. Math. Soc. 109 ( 1990 ) 157 - 164 . MR 1007505 | Zbl 0714.35029 · Zbl 0714.35029 [35] P. Lindqvist , Addendum to “On the equation div$$(|\nabla u|^{p-2}\nabla u)+ \Lambda |u|^{p-2}u$$ $$=0$$” . Proc. Amer. Math. Soc. 116 ( 1992 ) 583 - 584 . Zbl 0787.35027 · Zbl 0787.35027 [36] P. Lindqvist , Some remarkable sine and cosine functions . Ricerche Mat. 44 ( 1995 ) 269 - 290 . MR 1469702 | Zbl 0944.33002 · Zbl 0944.33002 [37] J.L. Lions , Quelques méthodes de résolutions des problèmes aux limites non linéaires . Dunod, Gauthier-Villars, Paris ( 1969 ). MR 259693 | Zbl 0189.40603 · Zbl 0189.40603 [38] M. Ohnuma and K. Sato , Singular degenerate parabolic equations with applications to the $$p$$-Laplace diffusion equation . Comm. Partial Differential Equations 22 ( 1997 ) 381 - 411 . MR 1443043 | Zbl 0990.35077 · Zbl 0990.35077 [39] M. Ôtani , Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations . J. Funct. Anal. 76 ( 1988 ) 140 - 159 . MR 923049 | Zbl 0662.35047 · Zbl 0662.35047 [40] S. Sakaguchi , Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems . Ann. Scuola Normale Superiore Pisa 14 ( 1987 ) 404 - 421 . Numdam | MR 951227 | Zbl 0665.35025 · Zbl 0665.35025 [41] G. Talenti , Personal Communication, letter dated Oct . 15, 2001 [42] P. Tolksdorf , Regularity for a more general class of quasilinear elliptic equations . J. Differential Equations 51 ( 1984 ) 126 - 150 . MR 727034 | Zbl 0488.35017 · Zbl 0488.35017 [42] N. Trudinger , On Harnack type inequalities and their application to quasilinear elliptic equations . Comm. Pure Appl. Math. 20 ( 1967 ) 721 - 747 . MR 226198 | Zbl 0153.42703 · Zbl 0153.42703 [43] N.N. Ural’tseva and A.B. Urdaletova , The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations . Vestnik Leningrad Univ. Math. 16 ( 1984 ) 263 - 270 . Zbl 0569.35029 · Zbl 0569.35029 [44] I.M. Višik , Sur la résolutions des problèmes aux limites pour des équations paraboliques quasi-linèaires d’ordre quelconque . Mat. Sbornik 59 ( 1962 ) 289 - 325 . [45] I.M. Višik , Quasilinear strongly elliptic systems of differential equations in divergence form . Trans. Moscow. Math. Soc. 12 ( 1963 ) 140 - 208 ; Translation from Tr. Mosk. Mat. Obs. 12 ( 1963 ) 125 - 184 . MR 156085 | Zbl 0144.36201 · Zbl 0144.36201
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