An integral estimate for the gradient for a class of nonlinear elliptic equations in the plane
The paper deals with the Dirichlet problem for second-order nonlinear differential equations \[ a(x,u,Du, D^2u)= f(x,u,Du)\quad\text{a.e. in }\Omega,\quad u= 0\quad\text{on }\partial\Omega,\tag{1} \] where \(\Omega\subset\mathbb{R}^2\) is a bounded and convex domain of class \(C^2\). It is supposed that \(a= a(x,z,p,\xi)\) and \(f= f(x,z,p)\) are real-valued functions satisfying the Carathéodory condition and \(a\) is supposed to satisfy the ellipticity condition of Campanato type, i.e., there exist constants \(\alpha\), \(\gamma\), \(\delta>0\), \(\gamma+\delta< 1\) such that \[ |\text{Tr}(\xi)- \alpha[a(x, z,p,\xi+ \tau)- a(x, z,p,\tau)]\leq \gamma\|\xi\|+ \delta|\text{Tr}(\xi)| \] for a.a. \(x\in\Omega\), all \(z\in\mathbb{R}\), \(p\in\mathbb{R}^2\), \(\xi,\tau\in \mathbb{R}^n\) and \(a(x, z,p,0)= 0\). Under certain conditions on the function \(f\) it is proved that there exists a constant \(C\) such that \(\| Du\|_{L^4(\Omega)}\leq C\) for each strong solution \(u\in W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)\) of (1). This gradient estimate combined with topological methods allows the author to prove strong solvability of the problem (1) in the class \(W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)\). The results of L. Softova are very interesting and certainly will contribute to obtain further new results in the theory of nonlinear elliptic equations.
scientific article; zbMATH DE number 1129189
scientific article