Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1132.35427
Birindelli, Isabeau; Demengel, Françoise
First eigenvalue and maximum principle for fully nonlinear singular operators. (English)
[J] Adv. Differ. Equ. 11, No. 1, 91-119 (2006). ISSN 1079-9389

The authors give a definition of the first eigenvalue for a class of fully nonlinear elliptic operators which are nonvariational but homogeneous. The main idea is to exploit the property that an elliptic operator satisfies a maximum principle if the involved parameter is less than the first eigenvalue. In the linear case it is well known that for the operator $Mu:= -\Delta u+\lambda u$ the maximum principle holds if $\lambda<\lambda_0$, where $\lambda_0$ is the first eigenvalue of $-\Delta u=\lambda u$ in $\Omega$, $u=0$ on $\partial\Omega$. Thus, $\lambda_0$ is the supremum of all $\lambda\in{\Bbb R}$ such that the maximum principle holds. The authors extend this view to a wide class of operators of the form $F(\nabla u,D^2 u)$. Let $\lambda_0$ be defined through the supremum of all $\lambda\in{\Bbb R}$ such that there exists $\Phi>0$ in $\Omega$ such that $F(\nabla\Phi,D^2\Phi)+\lambda\Phi^ {\alpha+1}\le0$ in the viscosity sense, where $\alpha>-1$. The authors show under additional assumptions that this $\lambda_0$ is the first eigenvalue of $-F$ in $\Omega$ in the sense that $F(\nabla u,D^2u)+\lambda|u|^\alpha u$ satisfies the maximum principle in $\Omega$ if $\lambda<\lambda_0$.
[Erich Miersemann (MR2192416)]

MSC 2000:: *35P30 Nonlinear eigenvalue problems for PD operators
35B50 Maximum principles (PDE)
35B65 Smoothness of solutions of PDE
35P15 Estimation of eigenvalues for PD operators

Cited in: Zbl 1223.35049 Zbl 1132.35032

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]