A free boundary problem for a singular system of differential equations: An application to a model of tumor growth

In this paper we consider a free boundary problem for a nonlinear system of two ordinary differential equations, one of which is singular at some points, including the initial point . Because of the singularity at , the initial value problem has a one-parameter family of solutions. We prove that there exists a unique solution to the free boundary problem. The proof of existence employs two ``shooting' parameters. Analysis of the profiles of solutions of the initial value problem and tools such as comparison theorems and weak limits of solutions play an important role in the proof. The system considered here is motivated by a model in tumor growth, but the methods developed should be applicable to more general systems.

     

Decay of global solutions, stability and blowup for a reaction-diffusion problem with free boundary

We consider a one-phase Stefan problem for the heat equation with a nonlinear reaction term. We first exhibit an energy condition, involving the initial data, under which the solution blows up in finite time in norm. We next prove that all global solutions are bounded and decay uniformly to 0, and that either: (i) the free boundary converges to a finite limit and the solution decays at an exponential rate, or (ii) the free boundary grows up to infinity and the decay rate is at most polynomial. Finally, we show that small data solutions behave like (i).

     

Multiplicity results for a class of superlinear elliptic problems

We study a class of superlinear elliptic problems under the Dirichlet boundary condition on a bounded smooth domain in . Assuming that the nonlinearity is superlinear in a neighborhood of , we study the dependence of the number of signed and sign-changing solutions on the parameter .

     

Stable and finite Morse index solutions on or on bounded domains with small diffusion

In this paper, we study bounded solutions of on (where and sometimes ) and show that, for most 's, the weakly stable and finite Morse index solutions are quite simple. We then use this to obtain a very good understanding of the stable and bounded Morse index solutions of on with Dirichlet or Neumann boundary conditions for small .

     

Global existence for energy critical waves in -d domains

We prove that the defocusing quintic wave equation, with Dirichlet boundary conditions, is globally well posed on for any smooth (compact) domain . The main ingredient in the proof is an spectral projector estimate, obtained recently by Smith and Sogge, combined with a precise study of the boundary value problem.

     

On a class of sublinear quasilinear elliptic problems

We establish existence and multiplicity of positive solutions to the quasilinear boundary value problem

where is a bounded domain in with smooth boundary , is continuous and p-sublinear at and is a large parameter.

     

Dynamical approach to some problems in integral geometry

As is well known, the main problem in integral geometry is to reconstruct a function in a given domain , where its integrals over a family of subdomains in are known. Such a problem is interesting not only as an object of pure analysis, but also in connection with various applications in practical disciplines. The most remarkable example of such a connection is the Radon problem and tomography. In this paper we solve one of these problems when is a bounded domain in with a piecewise smooth boundary. Some intermediate results related to dynamical systems with two generators and to some functional-integral equations are new and interesting per se. As an application of the results obtained we briefly study a boundary problem for a general third order hyperbolic partial differential equation in a bounded domain with data on the whole boundary .

     

The geometry of profinite graphs with applications to free groups and finite monoids

We initiate the study of the class of profinite graphs defined by the following geometric property: for any two vertices and of , there is a (unique) smallest connected profinite subgraph of containing them; such graphs are called tree-like. Profinite trees in the sense of Gildenhuys and Ribes are tree-like, but the converse is not true. A profinite group is then said to be dendral if it has a tree-like Cayley graph with respect to some generating set; a Bass-Serre type characterization of dendral groups is provided. Also, such groups (including free profinite groups) are shown to enjoy a certain small cancellation condition.

We define a pseudovariety of groups to be arboreous if all finitely generated free pro- groups are dendral (with respect to a free generating set). Our motivation for studying such pseudovarieties of groups is to answer several open questions in the theory of profinite topologies and the theory of finite monoids. We prove, for arboreous pseudovarieties , a pro- analog of the Ribes and Zalesski product theorem for the profinite topology on a free group. Also, arboreous pseudovarieties are characterized as precisely the solutions to the much studied pseudovariety equation .

     

Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations

We prove the uniqueness of Riemann solutions in the class of entropy solutions in for the system of compressible Euler equations, under usual assumptions on the equation of state for the pressure which imply strict hyperbolicity of the system and genuine nonlinearity of the first and third characteristic families. In particular, if the Riemann solutions consist of at most rarefaction waves and contact discontinuities, we show the global -stability of the Riemann solutions even in the class of entropy solutions in with arbitrarily large oscillation for the system. We apply our framework established earlier to show that the uniqueness of Riemann solutions implies their inviscid asymptotic stability under perturbation of the Riemann initial data, as long as the corresponding solutions are in and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any particular method for constructing the entropy solutions is made. Our uniqueness result for Riemann solutions can easily be extended to entropy solutions , piecewise Lipschitz in , for any 0$">.

     

Index estimates for minimal surfaces and -convexity

We prove Morse index estimates for the area functional for minimal surfaces that are solutions to the free boundary problem in -convex domains in manifolds of nonnegative complex sectional curvature.

     

Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems

We consider an initial boundary value problem for a symmetrizable mixed hyperbolic-parabolic system of conservation laws with a small viscosity , When the boundary is noncharacteristic for both the viscous and the inviscid system, and the boundary condition dissipative, we show that converges to a solution of the inviscid system before the formation of shocks if the amplitude of the boundary layer is sufficiently small. This generalizes previous results obtained for invertible and the linear study of Serre and Zumbrun obtained for a pure Dirichlet's boundary condition.

     

Semilinear elliptic equations and fixed points

In this paper, we deal with a class of semilinear elliptic equations in a bounded domain , , with boundary. Using a new fixed point result of the Krasnoselskii type for the sum of two operators, an existence principle of strong solutions is proved. We give two examples where the nonlinearity can be critical.

     

On the total curvature of convex hypersurfaces in hyperbolic spaces

Let be two convex compact subsets of the hyperbolic space with smooth boundary. It is shown that the total curvature of the hypersurface is larger than the total curvature of .

     

Global Hölder regularity for discontinuous elliptic equations in the plane

-regularity up to the boundary is proved for solutions of boundary value problems for elliptic equations with discontinuous coefficients in the plane.

In particular, we deal with the Dirichlet boundary condition

where , 2$">, or with the following normal derivative boundary conditions:

where , 2$">, 0$"> and is the unit outward normal to the boundary .

     

Asymptotics for the nonlinear dissipative wave equation

We are interested in the asymptotic behaviour of global classical solutions to the initial-boundary value problem for the nonlinear dissipative wave equation in the whole space or the exterior domain outside a star-shaped obstacle. We shall treat the nonlinear dissipative term like , , 0)$"> and prove that the energy does not in general decay. Further, we can deduce that the classical solution is asymptotically free and the local energy decays at a certain rate as the time goes to infinity.

     

Embedded minimal disks: Proper versus nonproper--global versus local

We construct a sequence of compact embedded minimal disks in a ball in with boundaries in the boundary of the ball and where the curvatures blow up only at the center. The sequence converges to a limit which is not smooth and not proper. If instead the sequence of embedded disks had boundaries in a sequence of balls with radii tending to infinity, then we have shown previously that any limit must be smooth and proper.

     

Regularity of isoperimetric hypersurfaces in Riemannian manifolds

We add to the literature the well-known fact that an isoperimetric hypersurface of dimension at most six in a smooth Riemannian manifold is a smooth submanifold. If the metric is merely Lipschitz, then is still Hölder differentiable.

     

Ideals of the cohomology rings of Hilbert schemes and their applications

We study the ideals of the rational cohomology ring of the Hilbert scheme of points on a smooth projective surface . As an application, for a large class of smooth quasi-projective surfaces , we show that every cup product structure constant of is independent of ; moreover, we obtain two sets of ring generators for the cohomology ring .

Similar results are established for the Chen-Ruan orbifold cohomology ring of the symmetric product. In particular, we prove a ring isomorphism between and for a large class of smooth quasi-projective surfaces with numerically trivial canonical class.

     

Pseudofree -actions on surfaces

In this paper, we give a weak classification of locally linear pseudofree actions of the cyclic group of order on a surface, and prove the existence of such an action which cannot be realized as a smooth action on the standard smooth surface.