A remark on the regularity of the solutions of the \(p\)-Laplacian and its application to their finite element approximation
The authors show that solutions of the inhomogeneous, two-dimensional \(p\)-Laplace equation with (nonzero) boundary data have higher regularity than previously known. Specifically, if \(\Omega\) is a domain in \(\mathbb{R}^ 2\), if \(f\in L^ q\) for some \(q>2\), and if \(1<p\leq 2\), then any weak solution of \(-\text{div} (|\nabla u|^{p-2} \nabla u)=f\) in \(\Omega\), \(u=g\) on \(\partial\Omega\) has weak second derivatives in \(L^ 2\) in either of two cases: (1) \(\partial\Omega \in C^ 2\) and \(g\in H^ 2\), or (2) \(\Omega\) is convex and \(g\equiv 0\). An estimate for these derivatives is given, and this estimate is used to obtain error bounds in a finite element approximation to the problem. Reviewer's comment: The author mentions regularity results of \textit{E. DiBenedetto} [Nonlinear Anal., Theory Methods Appl. 7, 827-850 (1983; Zbl 0539.35027)] and the reviewer [Nonlinear Anal., Theory Methods Appl. 12, No. 11, 1203-1219 (1988; Zbl 0675.35042)] in the introduction, but does not discuss these results in Remark 2.1, where they are particularly relevant. For example, if \(g\in C^{1,\alpha}\cap H^ 2\) for some \(\alpha\in (0,1)\), the reviewer's results ensure that the gradient of the solution is globally bounded and this fact simplifies the proofs of Theorems 2.1 and 2.2.
inhomogeneous, two-dimensional \(p\)-Laplace equation
error bounds in a finite element approximation
scientific article; zbMATH DE number 484029