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

Sergio Segura de León, José Toledo (1999)

Publicacions Matemàtiques

In this note we give some summability results for entropy solutions of the nonlinear parabolic equation ut - div ap (x, ∇u) = f in ] 0,T [xΩ with initial datum in L1(Ω) and assuming Dirichlet's boundary condition, where ap(.,.) is a Carathéodory function satisfying the classical Leray-Lions hypotheses, f ∈ L1 (]0,T[xΩ) and Ω is a domain in RN. We find spaces of type Lr(0,T;Mq(Ω)) containing the entropy solution and its gradient. We also include some summability results when f = 0 and the p-Laplacian...

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

Andrea Dall'Aglio, Sergio 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 ⊂ ℝ 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 .

Regularity of stable solutions of p -Laplace equations through geometric Sobolev type inequalities

Daniele Castorina, Manel Sanchón (2015)

Journal of the European Mathematical Society

We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of – Δ p u = g ( u ) in a smooth bounded domain Ω ⊂ ℝ n . In particular, we obtain new L r and W 1 , r bounds for the extremal solution u ☆ when the domain is strictly convex. More precisely, we prove that u ☆ ∈ L ∞ ( Ω ) if n ≤ p + 2 and u ☆ ∈ L n p n - p - 2 ( Ω ) ∩ W 0 1 , p ( Ω ) if n > p + 2 .

Remarques sur l’observabilité pour l’équation de Laplace

Kim-Dang Phung (2003)

ESAIM: Control, Optimisation and Calculus of Variations

Nous quantifions la propriété de continuation unique pour le laplacien dans un domaine borné quand la condition aux bords est a priori inconnue. Nous établissons une estimation de dépen-dance de type logarithmique suivant la terminologie de John [5]. Les outils utilisés reposent sur les inégalités de Carleman et les techniques des travaux de Robbiano [8, 11]. Aussi, nous déterminons en application de l’inégalité d’observabilité obtenue un coût du contrôle approché pour un problème elliptique modèle....

Remarques sur l'observabilité pour l'équation de Laplace

Kim-Dang Phung (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the Laplace equation in a smooth bounded domain. We prove logarithmic estimates, in the sense of John [5] of solutions on a part of the boundary or of the domain without known boundary conditions. These results are established by employing Carleman estimates and techniques that we borrow from the works of Robbiano [8,11]. Also, we establish an estimate on the cost of an approximate control for an elliptic model equation.

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