Boundary regularity under generalized growth conditions
In this paper, the authors consider minimizers of the integral function \[ \int_{\Omega}\varphi\big(x,|Du|\big)\ dx, \] where the function $\varphi$ is quite flexible inasmuch as its growth is concerned. For example, as part of the authors' results, the case \[ \varphi(x,t)=t^p, \] which is the standard Dirichlet $p$-energy, the case \[ \varphi(x,t)=t^{p(x)}, \] which is the variable growth setting, and the case \[ \varphi(x,t)=t^p+a(x)t^q, \] which is the double phase setting, are included as special cases of the main result. It is, therefore, interesting that the authors' results possess this degree of generality. The main result of the paper states that if the complement of the set $\Omega$ is (in the sense of capacity) locally fat at a given boundary point $x_0\in\partial\Omega$, then that point must be regular, which means that the boundary value is attained in a pointwise sense, not only in a Sobolev sense. The paper seems to be clearly written, and so, it would be of interest to researchers working in the general area of regularity theory in the calculus of variations or PDEs.
scientific article; zbMATH DE number 7039932
scientific article