Journal
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
Volume 495, Issue 1, Pages -Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jmaa.2020.124724
Keywords
p-Laplacian; Infinity-Laplacian; Convergence of the gradients
Categories
Funding
- Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior (PNPD/CAPES-UnB-Brazil) [88887.363582/2019-00]
- CONICET grant PIP GI (Argentina) [11220150100036CO]
- UBACyT (Argentina) [20020160100155BA]
- [MTM2015-70227-P]
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In this study, it is proven that the gradients of solutions to the Dirichlet problem for -Delta(p)u(p) = f, with f > 0, converge strongly to the gradient of the limit function as p tends to infinity in every L-q space with 1 <= q < infinity. The sharpness of this convergence is demonstrated with a simple 1-dimensional example showing no convergence in L-infinity. Additionally, the same strong convergence is obtained within the support of nonnegative f, and similar results hold true for the eigenvalue problem for a certain class of domains, such as balls or stadiums.
In this paper we prove that the gradients of solutions to the Dirichlet problem for -Delta(p)u(p) = f, with f > 0, converge as p -> infinity strongly in every L-q with 1 <= q < infinity to the gradient of the limit function. This convergence is sharp since a simple example in 1-d shows that there is no convergence in L-infinity. For a nonnegative f we obtain the same strong convergence inside the support of f. The same kind of result also holds true for the eigenvalue problem for a suitable class of domains (as balls or stadiums). (C) 2020 Elsevier Inc. All rights reserved.
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