Continuity properties of minimizers of integral functionals in a limit case
The authors prove the continuity of the minimizers \(u\in w^{1,p}(\Omega)\) for an integral functional of the form \({\mathcal F}(u)=\int_\Omega f(x,u(x),Du(x))dx\), where \(\Omega\) is an open subset of \(\mathbb{R}^n\). The function \(f\) is assumed to be a Carathéodory function of the form \(f_0+f_1\), where \(f_0\) satisfies a polynomial growth condition in the second and the third variable, and \(|f_1(x,\eta,\xi)|\leq g(x)|\xi|\), where \(g\) is a positive function in the Lorentz space \(L^{{n\over p-1},{1\over p-1}}(\Omega)\). The results obtained are sharpened versions of known continuity results by \textit{E. De Giorgi} [Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat., III. Ser. 3, 25-43 (1957)] and the first author [Boll. Unione Mat. Ital., VII. Ser. B8, No. 2, 257-270 (1994; Zbl 0799.35056)] concerning weak solutions of non-homogeneous elliptic partial differential equations. The motivation for the paper is the study of the regularity properties of elliptic equations which are the Euler-Lagrange equations of integral functionals. As an application of the techniques developed, the authors prove the continuity of solutions of the nonlinear equation \(\text{div }A(x,u,Du)+H(x,u)=\text{div }h\), under the hypotheses that \(h\in L^{{n\over p-1},{1\over p-1}}(\Omega)\), \(p>1\), \(A\) and \(H\) satisfy suitable monotonicity and polynomial growth assumptions.
weak solutions of nonlinear elliptic equations
scientific article; zbMATH DE number 930238
scientific article