The Wiener test for quasilinear elliptic equations with non-standard growth conditions (Q2735876)

The criterion of the regularity of a boundary point for the quasilinear partial differential equation NEWLINE\[NEWLINELu= \sum_{i=1}^n \frac{\partial}{\partial x_i} \Biggl(|\nabla u|^{p(x)-2} \frac{\partial u}{\partial x_i} \Biggr)= 0, \tag{1}NEWLINE\]NEWLINE where \(p(x)\) is measurable in the bounded domain \(D\) in \(\mathbb{R}^n\) and \(1< p-1\leq p(x)\leq p_2< \infty\) is investigated. This criterion is a generalization of the so-called Wiener test for the Laplace equation. NEWLINENEWLINENEWLINEIt is supposed that the function \(p(x)\) satisfies the condition NEWLINE\[NEWLINE|p(x)- p(y)|\leq \frac{\text{const}} {\ln\frac{1}{|x-y|}}, \qquad |x-y|\leq 1/2. \tag{2}NEWLINE\]NEWLINE At first, the analogues of the weak Harnack inequality for the supersolutions of the equation (1) are proved. NEWLINENEWLINENEWLINEFor the test establishing the Wiener's generalized solution of the Dirichlet problem for equation (1) is defined. The construction is based on the maximum principle. The main result contains the necessary and sufficient condition for regularity of a boundary point. It is formulated in terms of defined in the paper \(V_p(x)\)-capacity compact subset from some open ball. The test of this condition is difficult too. Therefore, in the paper it is formulated a geometric condition of regularity of a boundary point. In condition (2) an estimate of the modulus of the continuity for the Wiener solution of the problem for equation (1) near the boundary point is established.