Boundary estimates for solutions to the two-phase parabolic obstacle problem

Estimates for the second-order derivatives of a solution to the two-phase parabolic obstacle problem are established. Similar results in the elliptic case were obtained by the authors in 2006. Bibliography: 4 titles. Translated from Problemy Matematicheskogo Analiza, No. 38, December 2008, pp. 3–10.     

Gradient estimates for positive solutions of the Laplacian with drift

Let be a complete Riemannian manifold of dimension without boundary and with Ricci curvature bounded below by where If is a vector field such that and on for some nonnegative constants and then we show that any positive solution of the equation satisfies the estimate

on , for all In particular, for the case when this estimate is advantageous for small values of and when it recovers the celebrated Liouville theorem of Yau (Comm. Pure Appl. Math. 28 (1975), 201-228).

     

Optimal size estimates for the inverse conductivity problem with one measurement

We prove upper and lower estimates on the measure of an inclusion in a conductor in terms of one pair of current and potential boundary measurements. The growth rates of such estimates are essentially best possible.

     

On regularizing-rate estimates for solutions to the heat flow with initial value problem

For a compact Riemannian manifold NRK without boundary, we establish the existence of strong solutions to the heat flow for harmonic maps from Rn to N, and the regularizing rate estimate of the strong solutions. Moreover, we obtain the analyticity in spatial variables of the solutions. The uniqueness of the mild solutions in C([0,T]; W1,n) is also considered in this paper.     

Numerical estimates for stability domains of Lagrangian solutions to the restricted three-body problem

The geometric parameters of stability domains of Lagrangian solutions to the classical restricted three-body problem are quantitatively estimated. It is shown that these domains are ellipsoid-like plane figures stretched along the tangent to the circle that passes through the Lagrangian triangle solutions. A heuristic algorithm is proposed for determining the maximum size of these domains of attraction.     

Gradient estimates for viscosity solutions of singular fully nonlinear elliptic equations

We establish Lipschitz regularity for solutions to a family of non-isotropic fully nonlinear partial differential equations of elliptic type. In general such a regularity is optimal. No sign constraint is imposed on the solution, thus limiting free boundaries may have two-phases. Our estimates are then employed in combination with fine regularizing techniques to prove existence of viscosity solutions to singular nonlinear PDEs.     

Remarks on convexity estimates for solutions of the torsion problem

In this paper, for the solution of the torsion problem about the equation ?u =-2 with homogeneous Dirichlet boundary conditions in a bounded convex domain in Rn, we find a superharmonic function which implies the strict concavity of u^1/2 and give some convexity estimates. It is a generalization of Makar-Limanov’s result(Makar-Limanov(1971)) and Ma-Shi-Ye’s result(Ma et al.(2012)).     

Two-sided estimates for solutions to the Cauchy problem for Wazewski linear differential systems with delay

The Cauchy problem is considered for Wazewski linear differential systems with finite delay. The right-hand sides of systems contain nonnegative matrices and diagonal matrices with negative diagonal entries. The initial data are nonnegative functions. The matrices in equations are such that the zero solution is asymptotically stable. Two-sided estimates for solutions to the Cauchy problem are constructed with the use of the method of monotone operators and the properties of nonsingular M-matrices. The estimates from below and above are zero and exponential functions with parameters determined by solutions to some auxiliary inequalities and equations. Some estimates for solutions to several particular problems are constructed.     

Gradient estimates for solutions to quasilinear elliptic equations with critical sobolev growth and hardy potential

This note is a continuation of the work[17].We study the following quasilinear elliptic equations(■)where 1 p N,0 ≤μ ((N-p)/p)~p and Q ∈ L~∞(R~N).Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity.     

Some norm estimates for the solutions of a nonlinear diffusion problem

Summary By means of Rellich's identity bounds for the spectrum of the nonlinear problem v+e ^v=0 are derived and certain norms for the solutions are estimated.

---

Zusammenfassung Mit Hilfe einer Rellichschen Identität werden Schranken für das Spektrum des nichtlinearen Problems v+e ^v=0 angegeben. Ferner werden gewisse Normen für die Lösungen abgeschätzt.

     

Energy estimates for solutions of the mixed problem for linear second-order hyperbolic equations

A mixed problem for a linear second-order hyperbolic equation with antidissipation inside the domain and dissipation on a part of the boundary is considered. It is proved that for certain relations between the antidissipation inside the domain and the dissipation on the part of the boundary, the energy of the system exponentially decreases, whereas for sufficiently large antidissipation inside the domain the boundary dissipation has no effect on the energy of the system; in this case the energy remains unbounded. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 483–488, April, 1996.     

Gradient and Hölder estimates for positive solutions of Pucci type equations

We present some estimates for positive viscosity solutions of a class of fully non-linear elliptic equations including the extremal Pucci equations, generalizing some results for linear equations recently established by Y.Y. Li and L. Nirenberg. To cite this article: I. Capuzzo Dolcetta, A. Vitolo, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

A priori estimates and existence of solutions to the prescribed centroaffine curvature problem

In this paper we study the prescribed centroaffine curvature problem in the Euclidean space ${\mathbb{R}}^{n+1}$. This problem is equivalent to solving a Monge–Ampère equation on the unit sphere. It corresponds to the critical case of the Blaschke–Santaló inequality. By approximation from the subcritical case, and using an obstruction condition and a blow-up analysis, we obtain sufficient conditions for the a priori estimates, and the existence of solutions up to a Lagrange multiplier.     

Weak solutions of the forward problem in EEG for different conductivity values

The potential distribution on the scalp produced by current sources in the brain can be measured by an EEG recorder. The relationship between these sources and the scalp potential distribution may be described by a well-known mathematical model where some simplifications are usually introduced. The head is modeled as a multicompartment nested set and the conductivity of the different tissues is approximated by a positive piecewise constant function. This simplified model is used to solve the forward problem (FP), i.e., to calculate the scalp potential for a current source configuration. In this work, we prove that the weak solutions of the FP are continuous with respect to the conductivity values, that is, the difference between the scalp potentials is small if the conductivity values are closed enough. We present numerical examples that illustrates this property.     

Approximate solutions and error estimates for a Stokes boundary value problem

The aim of this paper is the numerical treatment of a boundary value problem for the system of Stokes’ equations. For this we extend the method of approximate approximations to boundary value problems. This method was introduced by Maz’ya (DFG-Kolloquium des DFG-Forschungsschwerpunktes Randelementmethoden, 1991) and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the system of Stokes’ equations in two dimensions. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström’s method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.     

Gradient continuity estimates

We consider degenerate elliptic equations of p-Laplacean type

$$-{\rm{div}}\, (\gamma(x)|Du|^{p-2}Du)=\mu\,,$$

and give a sufficient condition for the continuity of Du in terms of a natural non-linear Wolff potential of the right-hand side measure μ. As a corollary we identify borderline condition for the continuity of Du in terms of the data: namely μ belongs to the Lorentz space L(n, 1/(p ? 1)), and γ(x) is a Dini continuous elliptic coefficient. This last result, together with pointwise gradient bounds via non-linear potentials, extends to the non homogeneous p-Laplacean system, thereby giving a positive answer in the vectorial case to a conjecture of Verbitsky. Continuity conditions related to the density of μ, or to the decay rate of its L ⁿ -norm on small balls, are identified as well as corollaries of the main non-linear potential criterium.