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Regularity for entropy solutions of parabolic p-Laplacian type equations.

Sergio Segura de LeónJosé Toledo — 1999

Publicacions Matemàtiques

In this note we give some summability results for entropy solutions of the nonlinear parabolic equation u - div a (x, ∇u) = f in ] 0,T [xΩ with initial datum in L(Ω) and assuming Dirichlet's boundary condition, where a(.,.) is a Carathéodory function satisfying the classical Leray-Lions hypotheses, f ∈ L (]0,T[xΩ) and Ω is a domain in R. We find spaces of type L(0,T;M(Ω)) containing the entropy solution and its gradient. We also include some summability results when f = 0 and the p-Laplacian equation...

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

David ArcoyaSergio 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...

Regularity of renormalized solutions to nonlinear elliptic equations away from the support of measure data

Andrea Dall'AglioSergio Segura de León — 2019

Czechoslovak Mathematical Journal

We prove boundedness and continuity for solutions to the Dirichlet problem for the equation - div ( a ( x , u ) ) = h ( x , u ) + μ , in Ω N , where the left-hand side is a Leray-Lions operator from W 0 1 , p ( Ω ) into W - 1 , p ' ( Ω ) with 1 < p < N , h ( x , s ) is a Carathéodory function which grows like | s | p - 1 and μ is a finite Radon measure. We prove that renormalized solutions, though not globally bounded, are Hölder-continuous far from the support of μ .

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