This paper is devoted to the study of a class of degenerateboundary value problems for semilinear second-order elliptic differential operators depending nonlinearly on $\nabla u$. The purpose of this paper is to prove existence and uniqueness theorems of solutions of semilinear elliptic boundary value problems, by making use of the super-subsolution method and the Leray-Schauder degree theory. We remark that the variational method approach would fail, since our boundary condition is degenerate.

In a domain $\Omega \subset \mathbb{R}^n $, consider a weak solution $u$ of the Navier-Stokes equations in the class $u \in L^{\infty}(0, T; L^n(\Omega))$. If $\limsup_{t\to t_*-0}\|u(t)\|_n^n -\|u(t_*)\|_n^n$ is small at each point of $t_*\in(0, T)$, then $u$ is regular on $\bar{\Omega}\times (0, T)$. As an application, we give a precise characterization of the singular time; i.e., we show that if a solution $u$ of the Navier-Stokes equations is initially smooth and loses its regularity at some later time $T_* < T$, then either $\limsup_{t\to T_*-0}\|u(t)\|_{L^n(\Omega)} =+ \infty$, or $u(t)$ oscillates in $L^n(\Omega)$ around the weak limit w-$\lim_{t\to T_*-0}u(t)$ with sufficiently large amplitude. Furthermore, we prove that every weak solution $u$ of bounded variation on $(0, T)$ with values in $L^n(\Omega)$ becomes regular.

The solvability of semilinear elliptic equations with nonlinearities in the critical growth range depends on the terms with lower-order growth. We generalize some known results to a wide class of lower-order terms and prove a multiplicity result in the left neighborhood of every eigenvalue of $-\Delta$ when the subcritical term is linear. The proofs are based on variational methods; to assure that the considered minimax levels lie in a suitable range, special classes of approximating functions having disjoint support with the Sobolev ``concentrating" functions are constructed.

We study the long-time behavior of solutions to a degenerate wave equation describing nonlinear vibration of a string. For a certain class of initial data, we obtain the exact lower estimate of the rate of decay and the asymptotic expansion of solutions.

In this paper we relate the set of the orbits of a second-order Lagrangian systems joining two points on an open set with convex boundary of a Riemannian manifold with the topological structure of the open set. Such relations are obtained by developing a Morse Theory for the action integral of the Lagrangian system. Because of the presence of the boundary, the action integral does not satisfy the Palais-Smale condition. We perturb the action integral with a family of smooth functionals, satisfying the Palais-Smale condition. The Morse Relations for the action integral are obtained as limit of the Morse Relations of the perturbing functionals. A relation between the Morse index and the energy of the orbits as critical points of the action integral is obtained.

It is shown that surface tension effects on the free boundary are regularizing for Hele-Shaw models. This implies, in particular, existence and uniqueness of classical solutions for a large class of initial data. As a consequence, we give a rigorous proof of the fact that homogeneous Hele-Shaw flows with positive surface tension are volume preserving and area shrinking.

The Cauchy problem for linear constant--coefficient hyperbolic systems $u_t+Au_x=(1/\delta)Bu$ is analyzed. Here $(1/\delta)Bu$ is a large relaxation term, and we are mostly interested in the critical case where $B$ has a nontrivial nullspace. A concept of stiff well--posedness is introduced that ensures solution estimates independent of $0<\delta \leq 1$. Under suitable assumptions, we prove convergence of the $L_2$--solution to a limit as $\delta$ tends to zero. The limit solves a reduced strongly hyperbolic system without zero--order term, the so--called equilibrium system, and we present a method to determine this limit system. For 2$\times$2 systems the requirement of stiff well--posedness is shown to be equivalent to the well--known subcharacteristic condition, but in general the subcharacteristic condition is not sufficient for stiff well--posedness. The theory is illustrated by examples.

We establish the existence and partial regularity of a capacity solution to a coupled, degenerate, strongly nonlinear system of PDE's which models the melting of a solid due to volume electric heating. The system generalizes the usual Stefan problem, the evolutionary thermistor problem, and the spot welding problem. We allow temperature dependence for the electrical conductivity---which may lead to degeneracy---and take fully into account the flow of the fluid, which we model with the Navier-Stokes system. Existence is proved by considering a sequence of approximate problems, for which a priori estimates are obtained. Then the limit provides a capacity solution for the original problem. The approximate problems are obtained by smoothing, time-retardation and penalization. Of special interest is the fact that the set where the material is above its melting temperature is open, since only there the Navier-Stokes equations hold. The question of the behavior of solutions in mushy regions, regions where the temperature is identically the melting temperature, is left open.