We study a symmetric, nonlinear eigenvalue problem arising in earthquake initiation, and we establish the existence of infinitely many solutions. Under the effect of an arbitrary perturbation, we prove that the number of solutions becomes greater and greater if the perturbation tends to zero with respect to a prescribed topology. Our approach is based on nonsmooth critical-point theories in the sense of De Giorgi and Degiovanni.

Let $X$ be a locally compact space, $m$ a Radon measure on $X$,\, ${{\mathfrak h}}$ a regular Dirichlet form in $L_2(X,m)$. For a Radon measure $\mu$ we interpret ${{\mathfrak h}}$ as a regular Dirichlet form $\tau$ in $L_2(m+\mu)$. We show that $\mu$ decomposes as $\mu_r+\mu_s$, where $\mu_r$ is coupled to ${{\mathfrak h}}$ and $\mu_s$ decouples from ${{\mathfrak h}}$. Additionally to this `space perturbation', a second perturbation is introduced by a measure $\nu$ describing absorption. The main object of the paper is to apply this setting to a study of the Wentzell boundary condition \[ -\alpha Au + n\cdot a\nabla u + \gamma u = 0 \quad \text{on}\ \partial\Omega \] for an elliptic operator $A=-\nabla{\hspace{-0.1em}}\cdot(a\nabla)$, where $\Omega\subseteq{\mathbb{R}}^d$ is open, $n$ the outward normal, and $\alpha$, $\gamma$ are suitable functions. It turns out that the previous setting can be applied with $\mu=\alpha dS$, $\nu=\gamma dS$, under suitable conditions. Besides the description of the $d$-dimensional case we give a more detailed analysis of the one-dimensional case. As a further topic in the general setting we study the question whether mass conservation carries over from the unperturbed form ${{\mathfrak h}}$ to the space perturbed form $\tau$. In the Appendix we extend a known closability criterion from the minimal to the maximal form.

Let $(M,g)$ be a compact Riemannian manifold without boundary and dimension $d\ge3$. Let $u(t)$ be a solution to the problem $\dot u=\triangle_pu$, $u(0)=u_0$, $\triangle_p$ being the Riemannian $p$--Laplacian with $p>d$. Let also $\overline{u}$ be the (time--independent) mean of $u(t)$. We will prove ultracontractive estimates of the type $\Vert u(t)-\overline{u}\Vert_\infty\le C\Vert u(0)-\overline{u}\Vert_q^\gamma/t^\beta$. The constant $C$ depends only on $p$ and $q$, on geometric quantities of $M$ and on the dimension of the manifold, while the exponents $\beta$ and $\gamma$ depend only on $p$ and $q$ and differ according to the regimes $t\to0$ and $t\to+\infty$. Similar bounds hold when $\triangle_p$ is replaced by the subelliptic $p$--Laplacian associated to a collection of Hörmander vector fields. We also prove the L$^q$--L$^\infty$ Höolder continuity of the solutions, and apply similar methods to study the same questions for evolution equations on manifolds with boundary. The bounds are sharp in several of the above cases. The method relies on the theory of nonlinear Markov semigroups (\cite{CG2}) and on the connection between nonlinear ultracontractivity and logarithmic Sobolev inequality for the $p$--energy functional.

In this paper, we consider a system of two wave equations of Kirchhoff with coupled nonlinear and memory conditions at the boundary, and we study the asymptotic behavior of the corresponding solutions. We prove that the energy decays with the same rate of decay of the relaxation functions; that is, the energy decays exponentially when the relaxation functions decay exponentially and polynomially when the relaxation functions decay polynomially.