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Register a new userSolvability of Dirichlet problem with Integro-differential Operator
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This paper studies the solvability of a class of Dirichlet problem associated with non-linear integro-differential operator. The main ingredient is the probabilistic construction of continuous supersolution via the identification of the continuity set of the exit time operators under Skorohod topology.
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We consider stochastic differential equations driven by a general Levy processes (SDEs) with infinite activity and the related, via the Feynman-Kac formula, Dirichlet problem for parabolic integro-differential equation (PIDE). We approximate the solution of PIDE using a numerical method for the SDEs. The method is based on three ingredients: (i) we approximate small jumps by a diffusion; (ii) we use restricted jump-adaptive time-stepping; and (iii) between the jumps we exploit a weak Euler approximation. We prove weak convergence of the considered algorithm and present an in-depth analysis of how its error and computational cost depend on the jump activity level. Results of some numerical experiments, including pricing of barrier basket currency options, are presented.
We study acceleration phenomena in monostable integro-differential equations with ignition nonlinearity. Our results cover fractional Laplace operators and standard convolutions in a unified way, which is also a contribution of this paper. To achieve this, we construct a sub-solution that captures the expected dynamics of the accelerating solution, and this is here the main difficulty. This study involves the flattening effect occurring in accelerated propagation phenomena.
We solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in euclidean space. The main results apply, in particular, to subequations with a Riesz characteristic $p geq 2$. In this case it is shown that, without requiring uniform ellipticity, the Dirichlet problem can be solved uniquely for arbitrary continuous boundary data with singularities asymptotic to the Riesz kernel: $Theta_j K_p(x - x_j)$, where $K_p(x) = - {1over|x|^{p-2}}$ for $p>2$ and $K_2(x) = log |x|$, at any prescribed finite set of points $x_1,...,x_k$ in the domain and any finite set of positive real numbers $Theta_1,..., Theta_k$. This sharpens a previous result of the authors concerning the discreteness of high-density sets of subsolutions. Uniqueness and existence results are also established for finite-type singularities such as $Theta_j |x - x_j|^{2-p}$ for $1leq p<2$. The main results apply similarly with prescribed singularities asymptotic to the fundamental solutions of Armstrong-Sirakov-Smart (in the uniformly elliptic case).
A PDE system consisting of the momentum balance, mass balance, and energy balance equations for displacement, capillary pressure, and temperature as a model for unsaturated fluid flow in a porous viscoelastoplastic solid is shown to admit a solution under appropriate assumptions on the constitutive behavior. The problem involves two hysteresis operators accounting for plastic and capillary hysteresis.
In this paper, the regularity results for the integro-differential operators of the fractional Laplacian type by Caffarelli and Silvestre cite{CS1} are extended to those for the integro-differential operators associated with symmetric, regularly varying kernels at zero. In particular, we obtain the uniform Harnack inequality and Holder estimate of viscosity solutions to the nonlinear integro-differential equations associated with the kernels $K_{sigma, beta}$ satisfying $$ K_{sigma,beta}(y)asymp frac{ 2-sigma}{|y|^{n+sigma}}left( logfrac{2}{|y|^2}right)^{beta(2-sigma)}quad mbox{near zero} $$ with respect to $sigmain(0,2)$ close to $2$ (for a given $betainmathbb R$), where the regularity estimates do not blow up as the order $ sigmain(0,2)$ tends to $2.$
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