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Theoretical foundation of the weighted Laplace inpainting problem

Laurent Hoeltgen, Andreas Kleefeld, Isaac Harris, Michael Breuss (2019)

Applications of Mathematics

Laplace interpolation is a popular approach in image inpainting using partial differential equations. The classic approach considers the Laplace equation with mixed boundary conditions. Recently a more general formulation has been proposed, where the differential operator consists of a point-wise convex combination of the Laplacian and the known image data. We provide the first detailed analysis on existence and uniqueness of solutions for the arising mixed boundary value problem. Our approach considers...

Two-weight Sobolev-Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators

Bruno Franchi, Cristian E. Gutiérrez, Richard L. Wheeden (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this Note we prove a two-weight Sobolev-Poincaré inequality for the function spaces associated with a Grushin type operator. Conditions on the weights are formulated in terms of a strong A ∞ weight with respect to the metric associated with the operator. Roughly speaking, the strong A ∞ condition provides relationships between line and solid integrals of the weight. Then, this result is applied in order to prove Harnack's inequality for positive weak solutions of some degenerate elliptic equations....

Uniqueness of solutions for some degenerate nonlinear elliptic equations

Albo Carlos Cavalheiro (2014)

Applicationes Mathematicae

We investigate the existence and uniqueness of solutions to the Dirichlet problem for a degenerate nonlinear elliptic equation - ∑ i , j = 1 n D j ( a i j ( x ) D i u ( x ) ) + b ( x ) u ( x ) + d i v ( Φ ( u ( x ) ) ) = g ( x ) - ∑ j = 1 n f j ( x ) on Ω in the setting of the space H₀(Ω).

Uniqueness of solutions for some elliptic equations with a quadratic gradient term

David Arcoya, Sergio Segura de León (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We study a comparison principle and uniqueness of positive solutions for the homogeneous Dirichlet boundary value problem associated to quasi-linear elliptic equations with lower order terms. A model example is given by - Δ u + λ | ∇ u | 2 u r = f ( x ) , λ , r > 0 . The main feature of these equations consists in having a quadratic gradient term in which singularities are allowed. The arguments employed here also work to deal with equations having lack of ellipticity or some dependence on u in the right hand side. Furthermore, they...

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Erika Battaglia, Stefano Biagi, Andrea Bonfiglioli (0)

Annales de l’institut Fourier

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