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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 .

Currently displaying 21 – 40 of 56