Optimal regularity for the pseudo infinity Laplacian. (English) Zbl 1129.35087
The paper deals with optimal regularity for the viscosity solutions of the pseudo infinity Laplace equation given by
\[
\widetilde\Delta_\infty = \sum_{i\in I(\nabla u)} u_{x_ix_i}| u_{x_i}| ^2,
\]
where the summation is taken over the indices in \(I(\nabla u)=\{i: | u_{x_i}| =\max_j | u_{x_j}| \}.\) Local Lipschitz continuity is proved for the solutions and, by means of an explicit example, it is shown that this result is optimal. Existence and uniqueness of the respective Dirichlet problem are given as well.
Reviewer: Dian K. Palagachev (Bari)
MSC:
| 35R50 | PDEs of infinite order |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |
| 35J15 | Second-order elliptic equations |
| 35J60 | Nonlinear elliptic equations |
References:
| [1] | G. Aronsson , Extensions of functions satisfiying Lipschitz conditions . Ark. Math. 6 ( 1967 ) 551 - 561 . Zbl 0158.05001 · Zbl 0158.05001 · doi:10.1007/BF02591928 |
| [2] | G. Aronsson , M.G. Crandall and P. Juutinen , A tour of the theory of absolutely minimizing functions . Bull. Amer. Math. Soc. 41 ( 2004 ) 439 - 505 . Zbl pre02108961 · Zbl 1150.35047 · doi:10.1090/S0273-0979-04-01035-3 |
| [3] | G. Barles and J. Busca , Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term . Comm. Part. Diff. Eq. 26 ( 2001 ) 2323 - 2337 . Zbl 0997.35023 · Zbl 0997.35023 · doi:10.1081/PDE-100107824 |
| [4] | M. Belloni and B. Kawohl , The pseudo-p-Laplace eigenvalue problem and viscosity solutions as \(p\rightarrow \infty \) . ESAIM: COCV 10 ( 2004 ) 28 - 52 . Numdam | Zbl 1092.35074 · Zbl 1092.35074 · doi:10.1051/cocv:2003035 |
| [5] | M. Belloni , B. Kawohl and P. Juutinen , The p-Laplace eigenvalue problem as \(p\rightarrow \infty \) in a Finsler metric . J. Europ. Math. Soc. (to appear). Zbl pre05042376 · Zbl 1245.35078 |
| [6] | G. Bouchitte , G. Buttazzo and L. De Pasquale , A \(p-\)laplacian approximation for some mass optimization problems . J. Optim. Theory Appl. 118 ( 2003 ) 125. MR 1995692 | Zbl 1040.49040 · Zbl 1040.49040 · doi:10.1023/A:1024751022715 |
| [7] | 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. 27 ( 1992 ) 1 - 67 . arXiv | Zbl 0755.35015 · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5 |
| [8] | M.G. Crandall , L.C. Evans and R.F. Gariepy , Optimal Lipschitz extensions and the infinity Laplacian . Calc. Var. PDE 13 ( 2001 ) 123 - 139 . Zbl 0996.49019 · Zbl 0996.49019 · doi:10.1007/s005260000065 |
| [9] | L.C. Evans and W. Gangbo , Differential equations methods for the Monge-Kantorovich mass transfer problem . Mem. Amer. Math. Soc. 137 ( 1999 ), No. 653. MR 1464149 | Zbl 0920.49004 · Zbl 0920.49004 |
| [10] | R. Jensen , Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient . Arch. Rational Mech. Anal. 123 ( 1993 ) 51 - 74 . Zbl 0789.35008 · Zbl 0789.35008 · doi:10.1007/BF00386368 |
| [11] | O. Savin , \(C^1\) regularity for infinity harmonic functions in two dimensions . Arch. Rational Mech. Anal. 176 ( 2005 ) 351 - 361 . Zbl 1112.35070 · Zbl 1112.35070 · doi:10.1007/s00205-005-0355-8 |
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