We study two questions connected to the discrete spectrum of a three-dimensional \linebreak {\sch} operator with Coulomb potential perturbed by a spherically symmetric compactly supported function $q(r)$. It is known that the eigenvalues of this operator can be described by the formula $\lambda^\ell_n=-(n+\ell+\mu^\ell_n)^{-2}$, where $n=1,2,\dots$ is the principal quantum number, $\ell=0,1,2,\dots$ is the azimuthal quantum number (or angular momentum) and $\mu^\ell_n$ is the so-called quantum defect. It is, also, known that for each $\ell$ there exists a limit $\mu^\ell_\infty= {\lim_{n\rightarrow\infty}}\mu^\ell_n$. We assume that $q(r)\geq 0$ and prove the estimate $|\mu^\ell_\infty|\leq C(a e^2/2)^{2\ell+1} (\ell+0.5)^{-(4\ell+3)}$, where $a$ is the radius of the support of $q(r)$ and $C$ is $0.25 \sqrt{\pi}$ times the first moment of $q(r)$ (which is assumed to be finite). It follows from this estimate that $\mu^\ell_\infty$ tends to zero as $\ell\rightarrow \infty$ faster than any negative power of $\ell$. Our second result deals with the eigenfunctions $\{\Phi^\ell_n (r)\}^\infty_{n=1}$ of the radial {\sch} operator corresponding to a fixed value of $\ell$. The natural definition of $\Phi^\ell_n(r)$ which allows one to construct them explicitly, is based on the condition $ {\lim_{r\rightarrow 0}} r^{-\ell-1} \Phi^\ell_n(r)=1$. Such defined eigenfunctions are not normalized in $L^2(0,\infty)$ and the normalization constant $C^\ell_n$ is known only for the pure Coulomb potential. We prove that in the case of a perturbed Coulomb potential the constants $C^\ell_n$ behave as $n^{-3/2}$ when $n\rightarrow\infty$ for any fixed $\ell$ and, therefore, for any $\ell$ the system $\{n^{-3/2} \Phi^\ell_n (r)\}^\infty_{n=1}$ forms an orthogonal Riesz basis in its closed linear hull.

We study symmetry properties of positive solutions to some semilinear elliptic problems with nonlinear Neumann boundary conditions. We give sufficient conditions to have symmetry around the $\e_n$-axis of positive solutions of problems on the half-space. The proofs are based on the moving plane method. Finally some symmetry results are given in the case when the domain is a ball.

Consider (1.1) for a domain $\Omega$ for which there is no classical nonparametric solution of the stationary problem. We study viscosity solutions of (1.1). In general they fail to satisfy Dirichlet data on the boundary and "detach." In fact the solution tends to infinity with finite speed. The velocity stabilizes as $t\to \infty$, and we give some results on asymptotic growth. These new effects can be reconciled with the notion of viscosity solutions. The free boundary data are shown to be Lipschitzian for special domains $\Omega$. Problem (1.1) is related to some isoperimetric geometric problem.

This paper is concerned with the Hamilton-Jacobi equation of eikonal type $$ H(Du)=n(x) \qquad x\in \Omega \subset {\Bbb R}^N ,\tag E $$ where $H$ is convex, $Du$ represents the gradient of $u$ with respect to $x$, and $n(x)$ is lower semi-continuous. In this work, a new notion of generalized solution for (E) is developed which is appropriate for this class of discontinuous right-hand sides $n(x)$. Such solutions we term Monge solutions. The Monge notion arises in a natural way from the variational formulation of (E) and is consistent with the well-known viscosity notion when $n(x)$ is continuous. In the class of lower semi-continuous $n(x)$, we establish the comparison principle for Monge subsolutions and supersolutions, existence and uniqueness results for (E) with Dirichlet boundary conditions, and a stability result. Moreover, we show that the Monge solution can be smaller than the maximal Lipschitz subsolution.

It is shown that the solution of the Riemann problem $$ \hbox{$ \frac {\partial}{\partial t}$} \int_0^t k(t-s)(u(s,x)-u_0(x))\,\rm{d}s + (\sigma(u))_x(t,x) = 0, $$ where $u_0 = \chi_{\rminus}$ is continuous when $t> 0$. Here $k$ is locally integrable, nonnegative, and nonincreasing on $\mathbb{R}plus$ with $k(0+)=\infty$.

Consider the semilinear heat equation $u_{t}=\Delta u + u^{p}$, $t>0$, in an open set $\Omega \subset \mathbf {R}^N$, $p>1$. For every $a \in \Omega$, we construct solutions which blow up at $a$, at a finite time $T$, according to a variety of specific asymptotic behaviors. These blow-up patterns are unstable. The corresponding solutions have an arbitrary large number of local maxima, collapsing at $a$ for $t=T$.

In this paper we introduce a technique to construct new explicit solutions of nonlinear partial differential equations of mathematical physics with quadratic nonlinearities. We show that under certain assumptions the nonlinear operators under consideration admit so-called invariant sets on a two-dimensional linear functional subspace that makes it possible to rewrite the corresponding equation as an overdetermined finite-dimensional dynamical system. Some classes of such systems are shown to have nontrivial solutions. Examples of linear invariant subspaces for nonlinear quadratic operators from parabolic and hyperbolic equations are also given. In this case, equations on the corresponding invariant subspaces are reduced to finite-dimensional dynamical systems. Several generalizations including a similar analysis on three-dimensional subspaces, $N$-dimensional operators and equations with cubic nonlinearities are discussed.

We discuss the optimization problem of maximizing the torsional rigidity among bars with a given hole and a volume constraint. We find a class of holes for which the optimal bar exists and has a locally analytic outer boundary. We also prove that the problem is stable in this class.

We consider self-similar solutions of the fast diffusion equation $u_t=\nabla\cdot(u^{-n}\nabla u)$ in $(0,\infty)\times\mathbb{R}^N$, for $N\geq3$ and $\frac2N<n<1$, of the form $ u(x,t) = (T-t)^\alpha f\left(\left|x\right|(T-t)^{-\beta}\right). $ Because mass conservation does not hold for these values of $n$, this results in a nonlinear eigenvalue problem for $f$, $\alpha$ and $\beta$. We employ phase plane techniques to prove existence and uniqueness of solutions $(f,\alpha,\beta)$, and we investigate their behaviour when $n\uparrow1$ and when $n\downarrow\frac2N$.

We characterize the possible large time behaviors of solutions to $$ (|U|^{m-1} U)'' + {{N-1}\over\eta } (|U|^{m-1} U)' + \beta\eta U' + \alpha U + \gamma |U|^{p-1} U \, =\, 0 \quad \text{ in }\ \mathbb{R}^+ $$ with $m>0$, $p>1$, $\alpha, \beta >0$, $\alpha (m-1)+2\beta >0$ and $\gamma \in \{ -1,0,1\}$. It turns out that if $U$ is bounded and non-constant, $L:= \lim_{\eta\rightarrow\infty } \eta^{\alpha /\beta } U(\eta )$ is always finite and that in the case in which $L=0$, the solutions have compact support ($m>1$), decay exponentially ($m=1$) or decay like $\eta ^{-{2\over{1-m}}}$ ($m<1$), respectively. We want to stress that we impose no sign restriction on the solution.

We study the existence of multiple co-existence states of the Lotka-Volterra competition model in symmetric domains. We also investigate the stability of these states in various senses. Our method can also be used in the study of periodic solutions.

The initial value problem is studied for a parabolic variational inequality where the constraints are imposed on the time derivative of the solution and with a nonlinear right hand side. The existence of a solution is established under rather general conditions on the nonlinearity, while uniqueness and regularity properties are deduced in a more restrictive setting. An approximation via finite differences in time is presented and convergence results along with error estimates are proved. Applications of the model problem are given in the framework of nonlinear partial differential equations and systems.

The existence of solutions which are bounded on $ \mathbb{R}$, almost periodic or periodic is considered for a nonautonomous, singularly perturbed system of ordinary differential equations. In addition, the stability properties of these solutions are characterized by the construction of manifolds of initial data, the solutions for which approach the given solutions as $t\to+\infty$ ($t \to-\infty$) at an exponential rate, $\alpha$, independent of the small parameter. The key hypotheses are that certain linear systems have exponential dichotomies on $\mathbb{R}$. Applications are made to traveling wave solutions of reaction diffusion systems which are ``forced" by a traveling wave input.

We establish conditions sufficient to guarantee existence of nondecreasing solutions on $[0,1]$ of the differential equation $y''+f(t,y,y')=0$ subject to the boundary conditions $y(0)=0$, $y(1)=a>0$ or the initial conditions $y(0)=0$, $y'(0)=a>0$. Here $f$ is a nonnegative function which may be singular as $y\downarrow0$.

In this paper we consider periodic functional differential and integral equations. Massera proved that the existence of a bounded solution implies the existence of a periodic solution for linear (inhomogeneous) ordinary differential equations. Chow proved a similar result for linear functional differential equations with finite delay. We give a new proof for Chow's result that works for linear functional differential equations with infinite delay and also for integral equations. We also give examples that do not satisfy uniform boundedness or uniform ultimate boundedness, the conditions frequently used to prove that there is a periodic solution.

Inverse problems of dynamical identification of distributed control in hyperbolic variational inequalities are investigated. A numerical solution method based on constructions of the theory of positional control is described.