# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. (English) Zbl 1132.35032
The authors extend results of their previous paper [Adv. Differ. Equ. 11, No. 1, 91--119 (2006; Zbl 1132.35427)]. Precisely, define $$G(x,u,\nabla u, D^2u)=F(x,\nabla u, D^2u)+b(x)\cdot \nabla u\vert \nabla u\vert ^\alpha+c(x)\vert u\vert ^\alpha u,$$ where $F$ is continuous on $\Omega\times \bbfR^N\times S$, $S$ being the space of $N\times N$ symmetric matrices. Moreover, $F$ satisfies, for all real number $t$ and all non negative $\mu$ $$F(x,t p,\mu X)=\vert t\vert ^\alpha\mu F(x,p,X),$$ and for all $p\in \bbfR^N\setminus \{0\}$, $M\in S$, $N\in S$ with $N\ge 0$, $\alpha>-1$, $0<a\le A$ $$a\vert p\vert ^\alpha \mathrm{tr}(N)\le F(x,p,M+N)-F(x,p,M)\le A\vert p\vert ^\alpha \mathrm{tr}(N).$$ Moreover, suitable conditions about the uniform continuity of $F(x,p,X)$ are assumed. The functions $b:\Omega\to \bbfR^N$ and $c:\Omega\to \bbfR$, are supposed to be continuous and bounded. Additional conditions are assumed to prove specific theorems. The main object of the work is the following definition of eigenvalue $$\overline\lambda=\sup\{\lambda\in \bbfR:\exists \phi>0 \text{ in } \Omega,\ G(x,\phi,\nabla \phi,D^2\phi)+\lambda \phi^{1+\alpha}\le 0\},$$ where the inequality is understood in the viscosity sense. The following results are proved. (i) If $f\le 0$ is bounded and continuous in $\Omega$, and if $\lambda<\overline\lambda$ then there exists a nonnegative solution of the problem $$G(x,u,\nabla u, D^2u)+\lambda u^{1+\alpha}=f \text{ in }\ \Omega,\ u=0 \text{ on } \ \partial\Omega.$$ (ii) There exists $\phi>0$ in $\Omega$ such that $\phi$ is a viscosity solution of $$G(x,\phi,\nabla \phi, D^2\phi)+\lambda \phi^{1+\alpha}=f \text{ in }\ \Omega,\ \phi=0\ \text{ on }\ \partial\Omega.$$ (iii) $\phi$ is $\gamma$-Hölder continuous for all $\gamma\in (0,1)$ and locally Lipschitz continuous. To get the previous results, the authors prove a comparison principle and some boundary estimates which may have a own interest. Also, they prove regularity results which give the required relative compactness used to prove the main existence theorem.

##### MSC:
 35J60 Nonlinear elliptic equations 35P30 Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE) 35B50 Maximum principles (PDE) 35B65 Smoothness and regularity of solutions of PDE
##### Keywords:
fully nonlinear equation; viscosity solution; eigenvalue
Full Text: