Related papers: Fundamental solutions of homogeneous elliptic diff…
Differential constraints compatible with the linearized equations of partial differential equations are examined. Recursion operators are obtained by integrating the differential constraints.
In this paper we are concerned with the number of nonnegative solutions of the elliptic system $$ {array}{ll} -\Delta u = Q_u(u,v) + 1/2{2^*} H_u(u,v),& {in} \Omega,\vdois\ -\Delta v = Q_v(u,v) + 1/{2^*} H_v(u,v),& {in} \Omega,\vdois\…
We consider a class of first-order partial differential operators, acting on the space of ultradifferentiable periodic functions, and we describe their range by using the following conditions on the coefficients of the operators: the…
We prove H\"older regularity results for a class of nonlinear elliptic integro-differential operators with integration kernels whose ellipticity bounds are strongly directionally dependent. These results extend those in [9] and are also…
We propose a probabilistic definition of solutions of semilinear elliptic equations with (possibly nonlocal) operators associated with regular Dirichlet forms and with measure data. Using the theory of backward stochastic differential…
We consider the canonical fundamental systems of solutions of linear homogeneous Caputo fractional differential equations with continuous variable coefficients. Here we gained a series-representation of the canonical fundamental system by…
We give a simplified presentation of the obstacle problem approach to stochastic homogenization for elliptic equations in nondivergence form. Our argument also applies to equations which depend on the gradient of the unknown function. In…
We establish the existence of a bounded $H_\infty$-calculus for a large class of hypoelliptic pseudodifferential operators on R^n and closed manifolds.
In this work, the power series solutions are given around a regular-singular point, in the case of variable coefficients for homogeneous sequential linear conformable fractional differential equations of order 2{\alpha}.
We give an algorithm to compute inhomogeneous differential equations for definite integrals with parameters. The algorithm is based on the integration algorithm for $D$-modules by Oaku. Main tool in the algorithm is the Gr\"obner basis…
In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a selfadjoint matrix strongly elliptic second order differential operator $\mathcal{A}_\varepsilon$, $\varepsilon >0$. The coefficients of the operator $\mathcal{A}_\varepsilon$ are periodic…
This paper is concerned with Schr\"odinger equations whose principal operators are homogeneous elliptic. When the corresponding level hypersurface is convex, we show the $L^p$-$L^q$ estimate of solution operator in free case. This estimate,…
We study the regularity of the solution to an obstacle problem for a class of integro-differential operators. The differential part is a second order elliptic operator, whereas the nonlocal part is given by the integral fractional…
We present a new method for the existence and pointwise estimates of a Green's function of non-divergence form elliptic operator with Dini mean oscillation coefficients. We also present a sharp comparison with the corresponding Green's…
In this paper, we propose some algorithms for analytical solution construction to nonlinear polynomial partial differential equations with constant function coefficients. These schemes are based on one-(single), two- (double) or three-…
In this article we compare solutions to elliptic problems having rapidly oscillated conductivity (permeability, etc) coefficient with solutions to corresponding homogenized problems obtained from two-scale extensions of the initial…
In this short note, we investigate simultaneous recovery inverse problems for semilinear elliptic equations with partial data. The main technique is based on higher order linearization and monotonicity approaches. With these methods at…
We study the iterative algorithm proposed by S. Armstrong, A. Hannukainen, T. Kuusi, J.-C. Mourrat to solve elliptic equations in divergence form with stochastic stationary coefficients. Such equations display rapidly oscillating…
In this first part of the paper, we define a natural dual object for manifolds with corners and show how pseudodifferential calculus on such manifolds can be constructed in terms of the localization principle in C*-algebras. In the second…
A numerical method for variable coefficient elliptic problems on two dimensional domains is described. The method is based on high-order spectral approximations and is designed for problems with smooth solutions. The resulting system of…