Nonlinear parabolic functional differential equations with initial boundary conditions of the Neumann type are considered. A general class of difference methods for the problem is constructed. Theorems on the convergence of difference schemes and error estimates of approximate solutions are presented. The proof of the stability of the difference functional problem is based on a comparison technique. Nonlinear estimates of the Perron type with respect to the functional variable for given 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 $D\left(D_{n}u\right)\in L^p$.

We consider a stochastic Burgers equation. We show that the gradient of the corresponding transition semigroup $P_t\phi$ does exist for any bounded $\phi$; and can be estimated by a suitable exponential weight. An application to some Hamilton-Jacobi equation arising in Stochastic Control is given.

Let $\Omega$ be a bounded open subset of ${ℝ}^n$, let $X=(x,t)$ be a point of ${ℝ}^n\times {ℝ}^N$. In the cylinder $Q=\Omega \times (-T,0)$, $T>0$, we deduce the local differentiability result $$u\in L^2(-a,0,H^2(B\left(\sigma \right),{ℝ}^N))\cap H^1(-a,0,L^2(B\left(\sigma \right),{ℝ}^N))$$ for the solutions $u$ of the class $L^q(-T,0,H^{1,q}(\Omega ,{ℝ}^N))\cap C^{0,\lambda }(\overline{Q},{ℝ}^N)$ ($0<\lambda <1$, $N$ integer $\ge 1$) of the nonlinear parabolic system $$-\sum _{i=1}^n{D}_i{a}^i(X,u,Du)+{\displaystyle \frac{\partial u}{\partial t}}=B^0(X,u,Du)$$ with quadratic growth and nonlinearity $q\ge 2$. This result had been obtained making use of the interpolation theory and an imbedding theorem of Gagliardo-Nirenberg type for functions $u$ belonging to $W^{1,q}\cap C^{0,\lambda }$.