In this paper we prove a regularity result for very weak solutions of equations of the type $-divA(x,u,Du)=B(x,u,Du)$, where $A$, $B$ grow in the gradient like $t^{p-1}$ and $B(x,u,Du)$ is not in divergence form. Namely we prove that a very weak solution $u\in W^{1,r}$ of our equation belongs to $W^{1,p}$. We also prove global higher integrability for a very weak solution for the Dirichlet problem $$\left\{\begin{array}{cc}-divA(x,u,Du)=B(x,u,Du)\hfill & \text{in}\Omega ,u-u_o\in W^{1,r}(\Omega ,{ℝ}^m).\hfill \end{array}\right.$$

A priori estimates and strong solvability results in Sobolev space $W^{2,p}\left(\Omega \right)$, $1<p<\infty$ are proved for the regular oblique derivative problem $$\left\{\begin{array}{c}\sum _{i,j=1}^n{a}^{ij}\left(x\right)\frac{{\partial }^{2}u}{\partial x_i\partial x_j}=f\left(x\right)\text{a.e.}\Omega \hfill \\ \frac{\partial u}{\partial \ell }+\sigma \left(x\right)u=\varphi \left(x\right)\text{on}\partial \Omega \hfill \end{array}\right.$$ when the principal coefficients $a^{ij}$ are $VMO\cap L^{\infty }$ functions.

We consider a function $u:\Omega \to {ℝ}^N$, $\Omega \subset {ℝ}^n$, minimizing the integral ${\int }_{\Omega }\left(\right|D_1{u|}^2+\cdots +|D_{n-1}{u|}^2+|D_{n}u{|}^p)dx$, $2(n+1)/(n+3)\le p<2$, where $D_{i}u=\partial u/\partial x_i$, or some more general functional with the same behaviour; we prove the existence of second weak derivatives $D\left(D_{1}u\right),\cdots ,D\left(D_{n-1}u\right)\in L^2$ and $D\left(D_{n}u\right)\in L^p$.

Consider a nonlinear differential-functional equation (1) Au + f(x,u(x),u) = 0 where $Au:={\sum }_{i,j=1}^m{a}_{ij}\left(x\right)\left(\partial ²u\right)/\left(\partial x_i\partial x_j\right)$, $x=(x_1,...,x_m)\in G\subset {ℝ}^m$, G is a bounded domain with $C^{2+\alpha }$ (0 < α < 1) boundary, the operator A is strongly uniformly elliptic in G and u is a real $L^p\left(G̅\right)$ function. For the equation (1) we consider the Dirichlet problem with the boundary condition (2) u(x) = h(x) for x∈ ∂G. We use Chaplygin’s method [5] to prove that problem (1), (2) has at least one regular solution in a suitable class of functions. Using the method of upper...

In this paper, we shall be concerned with the existence result of the Degenerated unilateral problem associated to the equation of the type $Au+g(x,u,\nabla u)=f-\mathrm{div}F,$ where $A$ is a Leray-Lions operator and $g$ is a Carathéodory function having natural growth with respect to $\left|\nabla u\right|$ and satisfying the sign condition. The second term is such that, $f\in L^1\left(\Omega \right)$ and $F\in {\Pi }_{i=1}^N{L}^{p^{\prime }}(\Omega ,w_i^{1-p^{\prime }})$.