Convergence estimates of Galerkin-wavelet solutions to a Cauchy problem for a class of periodic pseudodifferential equations

A class of Cauchy problems for interesting complicated periodic pseudodifferential equations is considered. By the Galerkin-wavelet method and with weak solutions one can find sufficient conditions to establish convergence estimates of weak Galerkin-wavelet solutions to a Cauchy problem for this class of equations.

     

The Wiener-Hopf method for systems of pseudodifferential equations with an application to crack problems

The aim of this paper is to develop the Wiener-Hopf method for systems of pseudo-differential equations with non-constant coefficients and to apply it to the describtion of the asymptotic behaviour of solutions to boundary integral equations for crack problems when a crack occurs in a linear anisotropic elastic medium. The method was suggested in [15] for scalar pseudo-differential equations with constant coefficients and applied in [7] to the crack problems in the isotropic case. The existence and a-priori smoothness of solutions for the anisotropic case has been proved in [11, 12], while the isotropic case has been treated earlier in [7, 25, 41, 50]. Our results improve even those for the isotropic case obtained in [7, 50]. Asymptotic estimates for the behaviour of solutions in the anisotropic case have been obtained in [28] by a different method.In memoriam, dedicated to Professor Dr. V.D. Kupradze on the occasion of the 90th anniversary of his birthThis work was carried out during the first author's visit in Stuttgart in 1992 and supported by the DFG priority research programme Boundary Element Methods within the guest-programme We-659/19-2.     

Isoperimetric estimates for the first eigenfunction of a class of linear elliptic problems

We find some optimal estimates for the first eigenfunction of a class of elliptic equations whose prototype is - ( gu_xi )_xi = lgu \textin W ì \mathbbRⁿ - {\left( {\gamma u_{{x_{i} }} } \right)}_{{x_{i} }} = \lambda \gamma u\,{\text{in}}\,\Omega \subset \mathbb{R}^{n} with Dirichlet boundary condition, where γ is the normalized Gaussian function in \mathbbRⁿ \mathbb{R}^{n} . To this aim we make use of the Gaussian symmetrization which transforms a domain into an half-space with the same Gaussian measure. The main tools we use are the properties of the weighted rearrangements and in particular the isoperimetric inequality with respect to Gaussian measure.     

Isoperimetric estimates for the first eigenfunction of a class of linear elliptic problems

We find some optimal estimates for the first eigenfunction of a class of elliptic equations whose prototype is with Dirichlet boundary condition, where γ is the normalized Gaussian function in . To this aim we make use of the Gaussian symmetrization which transforms a domain into an half-space with the same Gaussian measure. The main tools we use are the properties of the weighted rearrangements and in particular the isoperimetric inequality with respect to Gaussian measure. Partially supported by GMAMPA - INDAM, Progetto “Proprietà analitico geometriche di soluzioni di equazioni ellittiche e paraboliche”.     

Euler solutions of pseudodifferential equations

We consider a homogeneous pseudodifferential equation on a cylinderC=×X over a smooth compact closed manifoldX whose symbol extends to a meromorphic function on the complex plane with values in the algebra of pseudodifferential operators overX. When assuming the symbol to be independent on the variablet , we show an explicit formula for solutions of the equation. Namely, to each non-bijectivity point of the symbol in the complex plane there corresponds a finite-dimensional space of solutions, every solution being the residue of a meromorphic form manufactured from the inverse symbol. In particular, for differential equations we recover Euler's theorem on the exponential solutions. Our setting is model for the analysis on manifolds with conical points sinceC can be thought of as a stretched manifold with conical points att=– andt=. When compared with the general theory, our approach is constructive while highlighting all the features of this latter.     

Global attractor and decay estimates of solutions to a class of nonlinear evolution equations

In this work, we prove the existence of global attractor for the nonlinear evolution equation u_tt?Δu?Δu_t?Δu_tt + g(x, u)=f(x) in X=(H²(Ω)∩H(Ω)) × (H²(Ω)∩H(Ω)). This improves a previous result of Xie and Zhong in (J. Math. Anal. Appl. 2007; 336 :54–69.) concerning the existence of global attractor in H(Ω) × H(Ω) for a similar equation. Further, the asymptotic behavior and the decay property of global solution are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

Convexity estimates for the solutions of a class of semi-linear elliptic equations

In this paper, we are concerned with convexity estimates for solutions of a class of semi-linear elliptic equations involving the Laplacian with power-type nonlinearities. We consider auxiliary curvature functions which attain their minimum values on the boundary and then establish lower bound convexity estimates for the solutions. Then we give two applications of these convexity estimates. We use the deformation method to prove a theorem concerning the strictly power concavity properties of the smooth solutions to these semi-linear elliptic equations. Finally, we give a sharp lower bound estimate of the Gaussian curvature for the solution surface of some specific equation by the curvatures of the domain's boundary.     

Capacitary estimates of solutions of a class of nonlinear elliptic equations

Let $\text{Ω}$ be a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$ and K a compact subset of $\text{?Ω}$. Assume that q?(N+1)/(N?1) and denote by U_K the maximal solution of ?Δu+u^q=0 in $\text{Ω}$ which vanishes on $\text{?Ω?K}$. We obtain sharp upper and lower estimates for U_K in terms of the Bessel capacity C_2/q,q′ and prove that U_K is σ-moderate. In addition we relate the strong ‘blow-up’ points of U_K on $\text{?Ω}$ to the ‘thick’ points of K in the fine topology associated with C_2/q,q′ and characterize these points by a path integral condition on U_K. To cite this article: M. Marcus, L. Véron, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Isoperimetric inequalities for a class of nonlinear parabolic equations

Abstrac Existence theorems and a priori bounds for a class of nonlinear parabolic equations are established. By means of an iteration process and symmetrization methods the solution in an arbitrary domain is compared with the one for the sphere of the same volume. It is shown that among all domains of given volume the sphere is the least stable.

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Zusammenfassung Mit Hilfe von Symmetrisierungen und Iterationsmethoden werden Existenzsätze und a priori Schranken für eine Klasse von nichtlinearen parabolischen Differentiagleichungen hergeleitet. Die Lösung für ein allgemeines Gebiet wird mit derjenigen für die Kugel vom gleichen Volumen verglichen. Es zeigt sich insbesondere, dass unter allen Gebieten mit demselben Volumen die Kugel am wenigsten stabil ist.

     

A priori estimates for solutions of boundary-value problems in a band for a class of higher-order degenerate elliptic equations

In this paper we consider two boundary-value problems in a band for higher-order degenerate elliptic equations. These equations degenerate on one boundary of the band to a third-order equation with respect to one variable. We study problems in weight spaces similar to Sobolev ones whose norms are constructed with the help of a certain integral transform. We obtain a priori estimates in these weight spaces for solutions to boundary-value problems in the band for higher-order elliptic equations that degenerate on one boundary of the band to a third-order equation with respect to one variable.     

Nonlocal time-multipoint problem for a class of evolution pseudodifferential equations

We prove the well-posed solvability of a nonlocal time-multipoint problem for evolution equations with pseudodifferential operators with analytic symbols and initial condition in the space of distributions of the type W′.     

Error estimates for series solutions to a class of nonlinear integral equations of mixed type

In this paper, we prove that the accelerated Adomian polynomials formula suggested by Adomian (Nonlinear Stochastic Systems: Theory and Applications to Physics, Kluwer, Dordrecht, 1989) and the accelerated formula suggested by El-Kalla (Int. J. Differ. Equs. Appl. 10(2):225?C234, 2005; Appl. Math. E-Notes 7:214?C221, 2007) are identically the same. The Kalla-iterates exhibit the same faster convergence exhibited by Adomian??s accelerated iterates with the additional advantage of absence of any derivative terms in the recursion, thereby allowing for ease of computation. Moreover, the formula of El-Kalla is used directly to prove the convergence of the series solution to a class of nonlinear two dimensional integral equations. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the Adomian series solution.