Boundary value problems for some fully nonlinear elliptic equations

We consider a Yamabe-type problem on locally conformally flat compact manifolds with boundary. The main technique we used is to derive boundary C ² estimates directly from boundary C ⁰ estimates. We will control the third derivatives on the boundary instead of constructing a barrier function. This result is a generalization of the work by Escobar.     

Boundary control problems for quasi-linear elliptic equations:

In this paper we prove some optimality conditions, in the form of a Pontryagin's principle, for boundary control problems governed by quasi-linear elliptic equations. Because of the presence of state constraints, we distinguish the cases of qualified and nonqualified conditions for optimality. Both cases are treated in the paper. Neither convexity of the control set nor differentiability of the functions involved in the control problem are assumed.This research was partially supported by Dirección General de Investigation Científica y Técnica (Madrid).     

Boundary value problems for quasi-linear elliptic second order equations in unbounded cone-like domains

We study the behaviour of weak solutions (as well as their gradients) of boundary value problems for quasi-linear elliptic divergence equations in domains extending to infinity along a cone.     

Boundary value problems for differential-algebraic equations

This paper shows how certain regularization methods can be used to determine the solution structure of linear differential systems subject to linear constraints and boundary conditions. Attention is restricted to the relatively straightforward index-one problems and to certain index-two problems, where endpoint jumps are related to the underlying boundary layer structure of the corresponding singular perturbation problems.     

Nonlinear nonuniformly elliptic second-order equations

One establishes a priori estimates for the first and second derivatives of the solutions of nonuniformly elliptic equations of the form F(x,u,Du,D^zu) without the assumptions of the convexity of the function F(x,z,p,r) with respect to r. These estimates allow us to extend the results of N. V. Krylov, L. C. Evans, and N. S. Trudinger on the classical solvability of the Dirichlet problem for essential nonlinear uniformly elliptic equations, convex with respect to D², to wider classes of nonlinear equations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 35–64, 1984.     

Parabolic equations with measurable coefficients II

We prove the existence and uniqueness of solutions in Sobolev spaces to second-order parabolic equations in non-divergence form. The coefficients (except one of them) of second-order terms of the equations are measurable in both time and one spatial variables, and VMO (vanishing mean oscillation) in other spatial variables.     

Boundary value problems for difference equations with causal operators

By using the monotone iterative method, some new results are established for nonlinear boundary conditions of difference problems with causal operators. We formulate sufficient conditions under which such problems have extremal solutions. Difference inequalities with causal operators are also discussed. Two examples are added to illustrate the results.     

Boundary value problems for differential equations with deviating arguments

Aequationes mathematicae -     

Boundary value problems for static Maxwell's equations

A generalized version of magnetostatics in differentiable manifolds is formulated. Different boundary value problems are treated as different representations of the same object as graphs of self-adjoint mappings. The Hodge theorem for a domain with the local segment property is proved.     

Boundary value problems for the Fitzhugh-Nagumo equations

This paper is concerned with the study of the FitzHugh-Nagumo equations. These equations arise in mathematical biology as a model of the transmission of electrical impulses through a nerve axon; they are a simplified version of the Hodgkin-Huxley equations. The FitzHugh-Nagumo equations consist of a non-linear diffusion equation coupled to an ordinary differential equation. v_t = v_xx + f(v) ? u, u_t = σv ? γu. We study these equations with either Dirichlet or Neumann boundary conditions, proving local and global existence, and uniqueness of the solutions. Furthermore, we obtain L_∞ estimates for the solutions in terms of the L₁ norm of the boundary data, when the boundary data vanish after a finite time and the initial data are zero. These estimates allow us to prove exponential decay of the solutions.