In this paper we prove that the gradients of solutions to the Dirichlet problem for −Δpup=f, with f>0, converge as p→∞ strongly in every Lq with 1≤q<∞ 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∞. 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).
Strong convergence of the gradients for p-Laplacian problems as p → ∞ / Buccheri, S.; Leonori, T.; Rossi, J. D.. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - 495:1(2020). [10.1016/j.jmaa.2020.124724]
Strong convergence of the gradients for p-Laplacian problems as p → ∞
Leonori T.
;
2020
Abstract
In this paper we prove that the gradients of solutions to the Dirichlet problem for −Δpup=f, with f>0, converge as p→∞ strongly in every Lq with 1≤q<∞ 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∞. 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).File | Dimensione | Formato | |
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