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July/August 2014 Existence and regularity for elliptic equations under $p,q$-growth
Giovanni Cupini, Paolo Marcellini, Elvira Mascolo
Adv. Differential Equations 19(7/8): 693-724 (July/August 2014).

Abstract

Under general $p,q$-growth conditions, we prove that the Dirichlet problem \begin{equation*} \left\{ \begin{array}{ll} \displaystyle \sum_{i=1}^{n}\frac{\partial }{\partial x_{i}}a^{i}(x,Du)=b(x) & \quad \text{in}\,\Omega , \\ u=u_{0} & \quad \text{on}\,\partial \Omega \end{array} \right. \end{equation*} has a weak solution $u\in W_{\mathrm{loc}}^{1,q} \Big ( \Omega \Big ) $ under the assumptions $ 1 < p\leq q\leq p+1$ and $q < p\tfrac{n-1}{n-p}. $ More regularity applies. Precisely, this solution is also in the class $W_{ \text{loc}}^{1,\infty }(\Omega )\cap {W_{\text{loc}}^{2,2}(\Omega )}$.

Citation

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Giovanni Cupini. Paolo Marcellini. Elvira Mascolo. "Existence and regularity for elliptic equations under $p,q$-growth." Adv. Differential Equations 19 (7/8) 693 - 724, July/August 2014.

Information

Published: July/August 2014
First available in Project Euclid: 6 May 2014

zbMATH: 1305.35041
MathSciNet: MR3252899

Subjects:
Primary: 35B65, 35J25

Rights: Copyright © 2014 Khayyam Publishing, Inc.

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Vol.19 • No. 7/8 • July/August 2014
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