Boundary decay estimates for solutions of fourth-order elliptic equations

We obtain integral boundary decay estimates for solutions of fourth-order elliptic equations on a bounded domain with regular boundary. We apply these estimates to obtain stability bounds for the corresponding eigenvalues under small perturbations of the boundary.     

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumptions of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma developed recently in Polá?ik et al. (2007) [20], we improve several Liouville-type theorems in higher order elliptic equations, some semilinear equations and elliptic systems. More specifically, we remove the boundedness assumption of the solutions which is required in the proofs of the corresponding Liouville-type theorems in the recent literature. Moreover, we also investigate the singularity and decay estimates of higher order elliptic equations.     

Uniform decay estimates for finite-energy solutions of semi-linear elliptic inequalities and geometric applications

We prove uniform decay estimates at infinity for solutions 0?u∈Lp of the semilinear elliptic inequality Δu+auσ+bu?0, a,b?0, σ?1, in the presence of a Sobolev inequality (with potential term). This gives a unified point of view in the investigation of different geometric questions. In particular, we present applications to the study of the topology at infinity of parallel mean curvature submanifolds, to the non-compact Yamabe problem, and to estimate the decay rate of the traceless Ricci tensor of conformally flat manifolds.     

A priori estimates for second order elliptic equations and their applications

We study the equation (λ+H)u=f whereH is a self-adjoint operator associated with the Dirichlet form inL ²(IR ^d ,pdx). A priori estimates of the first and the second order derivatives of solutions are obtained under minimal restrictions on the coefficients of the operator and measure. As a consequence we give a criterion of the essential self-adjointness of the operatorH ↾C ₀ ^∞ (IR ^d ) with non-smooth coefficients. Recipient of a Dov Biegun Postdoctoral Fellowship.     

Boundary blow-up solutions and their applications in quasilinear elliptic equations

Based on a comparison principle, we discuss the existence, uniqueness and asymptotic behaviour of various boundary blow-up solutions, for a class of quasilinear elliptic equations, which are then used to obtain a rather complete understanding of some quasilinear elliptic problems on a bounded domain or over the entireR ^N .     

On stationary zeros of solutions of linear elliptic equations

For a nontrivial solution of a linear homogeneous elliptic equation, we study the dimension of the set of zeros whose multiplicity is not less than the order of the equation. In the case of a linear homogeneous differential operator P = P(D) with constant coefficients and three variables, we show that if, for a solution of the equation Pu = 0, a point x ⁰ is a zero of multiplicity not less than the order of the equation, then the intersection of a sufficiently small neighborhood of the point x ⁰ with the set of all other zeros of this kind is a finite set of segments with common endpoint x ⁰.     

Priori estimates and asymptotic properties of solutions for some fractional order elliptic equations

In this paper, we obtain estimates for solutions for a class of fractional order elliptic equations in different domains and boundary conditions, and prove some regularity results. Then, we study the qualitative properties of solutions with prescribed Q-curvature.     

Exponential decay of solutions to nonlinear elliptic equations with potentials

Exponential decay estimates are obtained for complex-valued solutions to nonlinear elliptic equations in \mathbbRⁿ ,\mathbb{R}^{n} , where the linear term is given by Schr?dinger operators H =  − Δ  +  V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We describe specific rates of decay in terms of V, some of which are shown to be optimal. Moreover, our estimates provide a unified understanding of two distinct cases in the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V(x) = |x|².     

Exponential decay of solutions to nonlinear elliptic equations with potentials

Exponential decay estimates are obtained for complex-valued solutions to nonlinear elliptic equations in where the linear term is given by Schr?dinger operators H = − Δ + V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We describe specific rates of decay in terms of V, some of which are shown to be optimal. Moreover, our estimates provide a unified understanding of two distinct cases in the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V(x) = |x|². Dedicated to Professor Jun Uchiyama on the occasion of his sixtieth birthday Received: May 4, 2004     

New a priori estimates of solutions to anisotropic elliptic equations

Under consideration is the Dirichlet problem for singular anisotropic elliptic equations with a nonlinear source. Some new a priori estimates are obtained, implying that the solvability of the Dirichlet problem in the class of bounded solutions essentially depends on the dimension of the domain of the problem.     

Continuity properties of solutions to some degenerate elliptic equations

We consider a nonlinear (possibly) degenerate elliptic operator where the field a and the function b are (unnecessarily strictly) monotonic and a satisfies a very mild ellipticity assumption. For a given boundary datum ? we prove the existence of the maximum and the minimum of the solutions and formulate a Haar-Radò type result, namely a continuity property for these solutions that may follow from the continuity of ?. In the homogeneous case we formulate some generalizations of the Bounded Slope Condition and use them to obtain the Lipschitz or local Lipschitz regularity of solutions to Lu=0. We prove the global Hölder regularity of the solutions in the case where ? is Lipschitz.