For two general second order parabolic equations in divergence form in Lip(1,1/2) cylinders, we give a criterion for the preservation of $L^p$ solvability of the Dirichlet problems.

We prove the existence of the least positive eigenvalue with a corresponding nonnegative eigenfunction of the quasilinear eigenvalue problem $$\begin{array}{ccc}\hfill -div\left(a(x,u)\right|{|}^{p-2}\nabla u)=& {\lambda b(x,u)\left|u\right|}^{p-2}u\text{in}\Omega ,u=\hfill & \hfill 0\text{on}\partial \Omega ,\end{array}$$ where $\Omega$ is a bounded domain, $p>1$ is a real number and $a(x,u)$, $b(x,u)$ satisfy appropriate growth conditions. Moreover, the coefficient $a(x,u)$ contains a degeneration or a singularity. We work in a suitable weighted Sobolev space and prove the boundedness of the eigenfunction in $L^{\infty }\left(\Omega \right)$. The main tool is the investigation of the associated homogeneous eigenvalue problem and an application...

Inequalities concerning the integral of |∇u|2 on the subsets where |u(x)| is greater than k can be used in order to prove regularity properties of the function u. This method was introduced by Ennio De Giorgi e Guido Stampacchia for the study of the regularity of the solutions of Dirichlet problems.

We consider degenerated elliptic equations of the form$$\sum _{i,j}{D}_{x_i}(a_{ij}\left(x\right)D_{x_j}),\text{where}\lambda w(x\left)\right|\xi {|}^2\le \sum _{i,j}{a}_{ij}\left(x\right){\xi }_i{\xi }_j\le \Lambda w\left(x\right)|\xi {|}^2.$$Under suitable assumptions on $w$, we obtain a characterization of Wiener type (involving weighted capacities) for the set of regular points for these operators. The set of regular points is shown to depend only on $w$. The main tool we use is an estimate for the Green function in terms of $w$.

The spatial gradient of solutions to non-homogeneous and degenerate parabolic equations of $p$-Laplacean type can be pointwise estimated by natural Wolff potentials of the right hand side measure.

A parabolic system arisng as a viscosity regularization of the quasilinear one-dimensional telegraph equation is considered. The existence of $L\infty$ - a priori estimates, independent of viscosity, is shown. The results are achieved by means of generalized invariant regions.

We study boundary value problems for quasilinear parabolic equations when the initial condition is replaced by periodicity in the time variable. Our approach is to relate the theory of such problems to the classical theory for initial-boundary value problems. In the process, we generalize many previously known results.