On maps with given Jacobians involving the heat equation

In this paper we present and analyze two new algorithms to construct a smooth diffeomorphism of a domain with prescribed jacobian function. The first one is free from any restriction on the boundary, while the second one produces a diffeomorphism that coincides with the identity map on the boundary of the domain. Both are based on the solution of an initial value problem for the linear heat equation, and the second also uses solutions of the Stokes system of Fluid Mechanics.     

Multiplicative asymptotics of solutions of the first boundary value problem on a half-axis for a parabolic equation with a small parameter

In this work we consider the first boundary value problem for a parabolic equation of second order with a small parameter on a half-axis (i.e., we consider the one-dimensional case). We take the zero initial condition. We construct the global (that is, the caustic points are taken into account) asymptotics of a solution for the boundary value problem. The asymptotic solution of this problem has a different structure depending on the sign of the coefficient (the drift coefficient) at the derivative of first order at a boundary point. The constructed asymptotic solutions are justified.     

Convergence estimates of a projection-difference method for an operator-differential equation

This article investigates the projection-difference method for a Cauchy problem for a linear operator-differential equation with a leading self-adjoint operator A(t) and a subordinate linear operator K(t) in Hilbert space. This method leads to the solution of a system of linear algebraic equations on each time level; moreover, the projection subspaces are linear spans of eigenvectors of an operator similar to A(t). The convergence estimates are obtained. The application of the developed method for solving the initial boundary value problem is given.     

On Lundh’s percolation diffusion

A collection of spherical obstacles in the unit ball in Euclidean space is said to be avoidable for Brownian motion if there is a positive probability that Brownian motion diffusing from some point in the ball will avoid all the obstacles and reach the boundary of the ball. The centres of the spherical obstacles are generated according to a Poisson point process while the radius of an obstacle is a deterministic function. If avoidable configurations are generated with positive probability, Lundh calls this percolation diffusion. An integral condition for percolation diffusion is derived in terms of the intensity of the point process and the function that determines the radii of the obstacles.     

Green and Poisson functions with Wentzell boundary conditions

We discuss the construction and estimates of the Green and Poisson functions associated with a parabolic second order integro-differential operator with Wentzell boundary conditions.     

Multiple equilibria,periodic solutions and a priori bounds for solutions in superlinear parabolic problems

Consider the Dirichlet problem for the parabolic equation in , where $\Omega$ is a bounded domain in and f has superlinear subcritical growth in u. If f is independent of t and satisfies some additional conditions then using a dynamical method we find multiple (three, six or infinitely many) nontrivial stationary solutions. If f has the form where m is periodic, positive and m,g satisfy some technical conditions then we prove the existence of a positive periodic solution and we provide a locally uniform bound for all global solutions.     

Prescribed mean curvature equations under the transformation with non-orthogonal curvilinear coordinates

The aim of this paper is to investigate the Dirichlet problem of prescribed mean curvature equations. We show the existence of a weak solution. The boundary of domains does not always satisfy the H-convexity condition. Our method is not to construct the barrier functions directly, but to use some uniform estimate for solutions of the approximating regularized solutions.     

Three-term asymptotics for the boundary layers of semilinear elliptic eigenvalue problems

We consider the nonlinear eigenvalue problem −Δu=λ f(u) in Ω u=0 on ∂Ω, where Ω is a ball or an annulus in R^N (N ≥ 2) and λ > 0 is a parameter. It is known that if λ >> 1, then the corresponding positive solution u_λ develops boundary layers under some conditions on f. We establish the asymptotic formulas for the slope of the boundary layers of u_λ with the exact second term and the ‘optimal’ estimate of the third term.     

A minimum problem with free boundary in Orlicz spaces

We consider the optimization problem of minimizing in the class of functions W1,G(Ω) with , for a given φ₀?0 and bounded. W1,G(Ω) is the class of weakly differentiable functions with . The conditions on the function G allow for a different behavior at 0 and at ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω∩∂{u>0}, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the C1,α regularity of their free boundaries near “flat” free boundary points.     

Uniqueness for an elliptic-parabolic problem with Neumann boundary condition

We consider the problem in a smooth boundary domain , as well as the corresponding evolution equation . For the stationary equation we show existence results, then we adapt the techniques of doubling of variables to the case of the homogeneous Neumann boundary conditions and obtain the appropriate L ¹ -contraction principle and uniqueness. Subsequently, we are able to apply the nonlinear semigroup theory and prove the L ¹ -contraction principle for the associated evolution equation.     

Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space

We consider the entire graph S of a continuous real function over RN−1 with N?3. Let Ω be a domain in RN with S as a boundary. Consider in Ω the heat flow with initial temperature 0 and boundary temperature 1. The problem we consider is to characterize S in such a way that there exists a stationary isothermic surface in Ω. We show that S must be a hyperplane under some general conditions on S. This is related to Liouville or Bernstein-type theorems for some elliptic Monge-Ampère-type equation.     

Regularity of the extremal solution for some elliptic problems with singular nonlinearity and advection

In this note, we investigate the regularity of the extremal solution u^? for the semilinear elliptic equation −△u+c(x)⋅∇u=λf(u) on a bounded smooth domain of Rn with Dirichlet boundary condition. Here f is a positive nondecreasing convex function, exploding at a finite value a∈(0,∞). We show that the extremal solution is regular in the low-dimensional case. In particular, we prove that for the radial case, all extremal solutions are regular in dimension two.     

Existence and non-existence of solutions for a class of Monge-Ampère equations

We study the boundary value problems for Monge-Ampère equations: detD²u=e−u in Ω⊂Rn, n?1, u_|∂Ω=0. First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a solution, and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. Finally we consider an equivalent problem with a parameter detD²u=e−tu in Ω, u_|∂Ω=0, t?0. By using Lyapunov-Schmidt reduction method we get the local structure of the solutions near a degenerate point; by Leray-Schauder degree theory, a priori estimate and bifurcation theory we get the global structure.     

Heat flow, Brownian motion and Newtonian capacity

Let K be a compact, non-polar set in Rm(m?3) and let u be the unique weak solution of on Rm\K×(0,∞),u(x;0)=0 on Rm\K and u(x;t)=1 for all x on the boundary of K and for all t>0. The asymptotic behaviour of u(x;t) as t tends to infinity is obtained up to order O(t−m/2).     

Exterior nonlinear Neumann problem

We consider the solvability of the Neumann problem for equation (1.1) in exterior domains in both cases: subcritical and critical. We establish the existence of least energy solutions. In the subcritical case the coefficient b(x) is allowed to have a potential well whose steepness is controlled by a parameter λ > 0. We show that least energy solutions exhibit a tendency to concentrate to a solution of a nonlinear problem with mixed boundary value conditions.     

Large time behavior for a viscous Hamilton-Jacobi equation with Neumann boundary condition

We prove the existence and the uniqueness of strong solutions for the viscous Hamilton-Jacobi equation: with Neumann boundary condition, and initial data μ₀, a continuous function. The domain Ω is a bounded and convex open set with smooth boundary, a∈R,a≠0 and p>0. Then, we study the large time behavior of the solution and we show that for p∈(0,1), the extinction in finite time of the gradient of the solution occurs, while for p?1 the solution converges uniformly to a constant, as t→∞.     

Cross-shaped and degenerate singularities in an unstable elliptic free boundary problem

We investigate singular and degenerate behavior of solutions of the unstable free boundary problem

Δu=−χ{u>0}.     

On the Cauchy problem for integro-differential operators in Sobolev classes and the martingale problem

The existence and uniqueness in Sobolev spaces of solutions of the Cauchy problem to parabolic integro-differential equation with variable coefficients of the order α∈(0,2)$\alpha \in (0,2)$ is investigated. The principal part of the operator has kernel m(t,x,y)/|y|^d+α$m(t,x,y)/{|y|}^{d+\alpha }$ with a bounded nondegenerate m, Hölder in x and measurable in y. The lower order part has bounded and measurable coefficients. The result is applied to prove the existence and uniqueness of the corresponding martingale problem.     

Boundary behaviour for solutions of elliptic singular equations with a gradient term

We investigate the homogeneous Dirichlet boundary value problem for a class of second-order nonlinear elliptic partial differential equations with a gradient term and singular data. Under general conditions on the data, we study the behaviour of the solution near the boundary of the domain. Under suitable additional conditions we also investigate the second-order term in the asymptotic expansion of the solution in terms of the distance from the boundary.     

Some properties of solutions to the singular quasilinear problems

In this paper, a kind of quasilinear elliptic problem is studied, which involves the critical exponent and singular potentials. By the Caffarelli-Kohn-Nirenberg inequality and variational methods, some important properties of the positive solution to the problem are established.