Displaying similar documents to “Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations”

Existence and uniqueness results for solutions of nonlinear equations with right hand side in L 1

A. Fiorenza, C. Sbordone (1998)

Studia Mathematica

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We prove an existence and uniqueness theorem for the elliptic Dirichlet problem for the equation div a(x,∇u) = f in a planar domain Ω. Here f ∈ L 1 ( Ω ) and the solution belongs to the so-called grand Sobolev space W 0 1 , 2 ) ( Ω ) . This is the proper space when the right hand side is assumed to be only L 1 -integrable. In particular, we obtain the exponential integrability of the solution, which in the linear case was previously proved by Brezis-Merle and Chanillo-Li.

Capacitary strong type estimates in semilinear problems

D. Adams, Michel Pierre (1991)

Annales de l'institut Fourier

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We prove the equivalence of various capacitary strong type estimates. Some of them appear in the characterization of the measures μ that are admissible data for the existence of solutions to semilinear elliptic problems with power growth. Other estimates are known to characterize the measures μ for which the Sobolev space W 2 , p can be imbedded into L p ( μ ) . The motivation comes from the semilinear problems: simpler descriptions of admissible data are given. The proof surprisingly involves the...

On some nonlinear partial differential equations involving the 1-Laplacian

Mouna Kraïem (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let Ω be a smooth bounded domain in ℝ N , N > 1 and let n ∈ ℕ * . We prove here the existence of nonnegative solutions u n in B V ( Ω ) , to the problem ( P n ) - div σ + 2 n ∫ Ω u - 1 sign + ( u ) = 0 in Ω , σ · ∇ u = | ∇ u | in Ω , u is not identically zero , - σ · n → u = u on ∂ Ω , where n → denotes the unit outer normal to ∂ Ω , and sign + ( u ) denotes some L ∞ ( Ω ) function defined as: sign + ( u ) . u = u + , 0 ≤ sign + ( u ) ≤ 1 . Moreover, we prove the tight convergence of u n towards one of the first eingenfunctions for the first 1 - Laplacian Operator - Δ 1 on Ω when n goes to + ∞ .

Some simple nonlinear PDE's without solutions

Haïm Brezis, Xavier Cabré (1998)

Bollettino dell'Unione Matematica Italiana

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In questo articolo consideriamo alcune semplici equazioni a derivate parziali elittiche nonlineari, per le quali il Teorema della Funzione Inversa, se applicato in modo formale, suggerisce l'esistenza di soluzioni. Nonostante ciò, proviamo che non esistono soluzioni neppure in vari sensi deboli. Un problema modello è dato da - Δ ⁢ u = u 2 / x 2 + c in Ω , u = 0 su ∂ ⁡ Ω , dove Ω ⊂ R N , N ≥ 2 , è un dominio limitato contenente 0 . Per qualunque costante c > 0 , arbitrariamente piccola, proviamo che questo problema non ammette soluzioni...

Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations

Francesca Da Lio, Boyan Sirakov (2007)

Journal of the European Mathematical Society

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We study uniformly elliptic fully nonlinear equations F ( D 2 u , D u , u , x ) = 0 , and prove results of Gidas–Ni–Nirenberg type for positive viscosity solutions of such equations. We show that symmetries of the equation and the domain are reflected by the solution, both in bounded and unbounded domains.