Searcharxiv⌕ Search

arXiv subjects

Erwin Topp

Publications and source records attributed to Erwin Topp.

30 records · Page 2Linked to original sources
arXiv2020

Interior regularity results for fractional elliptic equations that degenerate with the gradient

In this paper we obtain interior regularity estimates for viscosity solutions of nonlocal Dirichlet problems that degenerate when the gradient of the solution vanishes. Interior Hölder estimates are obtained when the order of the fractional diffusion is less or equal than one, and Lipschitz estimates when it is bigger than one. In the latter case, the estimates are robust enough to conclude interior $C^{1, α}$ regularity by an improvement of the flatness procedure, which is possible when the nonlocal term is close enough to a second-order diffusion.

math.AP↗
arXiv2020

Periodic Homogenization for Weakly Elliptic Hamilton-Jacobi-Bellman Equations with Critical Fractional Diffusion

In this paper we establish periodic homogenization for Hamilton-Jacobi-Bellman (HJB) equations, associated to nonlocal operators of integro-differential type. We consider the case when the fractional diffusion has the same order as the drift term, and is weakly elliptic. The outcome of the paper is twofold. One one hand, we provide Lipschitz regularity results for weakly elliptic non-local HJB, extending the results previously obtained in [8]. On the other hand, we establish a convergence result, based on half relaxed limits and a comparison principle for the effective problem. The latter strongly relies on the regularity and the ellipticity properties of the effective Hamiltonian, for which a fine Lipschitz estimate of the corrector plays a crucial role.

math.AP↗
arXiv2019

Some results for the large time behavior of Hamilton-Jacobi Equations with Caputo Time Derivative

We obtain some Hölder regularity estimates for an Hamilton-Jacobi with fractional time derivative of order $α\in (0,1)$ cast by a Caputo derivative. The Hölder seminorms are independent of time, which allows to investigate the large time behavior of the solutions. We focus on the Namah-Roquejoffre setting whose typical example is the Eikonal equation. Contrary to the classical time derivative case $α=1$, the convergence of the solution on the so-called projected Aubry set, which is an important step to catch the large time behavior, is not straightforward. Indeed, a function with nonpositive Caputo derivative for all time does not necessarily converge; we provide such a counterexample. However, we establish partial results of convergence under some geometrical assumptions.

math.AP↗
arXiv2019

Cauchy problem and periodic homogenization for nonlocal Hamilton-Jacobi equations with coercive gradient terms

This paper deals with the periodic homogenization of nonlocal parabolic Hamilton-Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal operator, giving rise to three different limit problems. To prove the locally uniform convergence to the unique solution of the Cauchy problem for the effective equation we need a new comparison principle among viscosity semi-solutions of integro-differential equations that can be of independent interest.

math.AP↗
arXiv2016

Lipschitz regularity for integro-differential equations with coercive hamiltonians and application to large time behavior

In this paper, we provide suitable adaptations of the "weak version of Bernstein method" introduced by the first author in 1991, in order to obtain Lipschitz regularity results and Lipschitz estimates for nonlinear integro-differential elliptic and parabolic equations set in the whole space. Our interest is to obtain such Lipschitz results to possibly degenerate equations, or to equations which are indeed "uniformly el-liptic" (maybe in the nonlocal sense) but which do not satisfy the usual "growth condition" on the gradient term allowing to use (for example) the Ishii-Lions' method. We treat the case of a model equation with a superlinear coercivity on the gradient term which has a leading role in the equation. This regularity result together with comparison principle provided for the problem allow to obtain the ergodic large time behavior of the evolution problem in the periodic setting.

math.AP↗
arXiv2015

Lipschitz Regularity for Censored Subdiffusive Integro-Differential Equations with Superfractional Gradient Terms

In this paper we are interested in integro-differential elliptic and parabolic equations involving nonlocal operators with order less than one, and a gradient term whose coercivity growth makes it the leading term in the equation. We obtain Lipschitz regularity results for the associated stationary Dirichlet problem in the case when the nonlocality of the operator is confined to the domain, feature which is known in the literature as censored nonlocality. As an application of this result, we obtain strong comparison principles which allow us to prove the well-posedness of both the stationary and evolution problems, and steady/ergodic large time behavior for the associated evolution problem.

math.AP↗
arXiv2014

Uniform Equicontinuity for a family of Zero Order operators approaching the fractional Laplacian

In this paper we consider a smooth bounded domain $Ω\subset \R^N$ and a parametric family of radially symmetric kernels $K_ε: \R^N \to \R_+$ such that, for each $ε\in (0,1)$, its $L^1-$norm is finite but it blows up as $ε\to 0$. Our aim is to establish an $ε$ independent modulus of continuity in $Ω$, for the solution $u_ε$ of the homogeneous Dirichlet problem \begin{equation*} \left \{ \begin{array}{rcll} - \I_ε[u] \&=\& f \& \mbox{in} \ Ω. \\ u \&=\& 0 \& \mbox{in} \ Ω^c, \end{array} \right . \end{equation*} where $f \in C(\barΩ)$ and the operator $\I_ε$ has the form \begin{equation*} \I_ε[u](x) = \frac12\int \limits_{\R^N} [u(x + z) + u(x - z) - 2u(x)]K_ε(z)dz \end{equation*} and it approaches the fractional Laplacian as $ε\to 0$. The modulus of continuity is obtained combining the comparison principle with the translation invariance of $\I_ε$, constructing suitable barriers that allow to manage the discontinuities that the solution $u_ε$ may have on $\partial Ω$. Extensions of this result to fully non-linear elliptic and parabolic operators are also discussed.

math.AP↗
arXiv2014

Existence, Uniqueness and Asymptotic Behavior for Nonlocal Parabolic Problems with Dominating Gradient Terms

In this paper we deal with the well-posedness of Dirichlet problems associated to nonlocal Hamilton-Jacobi parabolic equations in a bounded, smooth domain $Ω$, in the case when the classical boundary condition may be lost. We address the problem for both coercive and noncoercive Hamiltonians: for coercive Hamiltonians, our results rely more on the regularity properties of the solutions, while noncoercive case are related to optimal control problems and the arguments are based on a careful study of the dynamics near the boundary of the domain. Comparison principles for bounded sub and supersolutions are obtained in the context of viscosity solutions with generalized boundary conditions, and consequently we obtain the existence and uniqueness of solutions in $C(\barΩ \times [0,+\infty))$ by the application of Perron's method. Finally, we prove that the solution of these problems converges to the solutions of the associated stationary problem as $t \to +\infty$ under suitable assumptions on the data.

math.AP↗
arXiv2014

Regularity Results and Large Time Behavior for Integro-Differential Equations with Coercive Hamiltonians

In this paper we obtain regularity results for elliptic integro-differential equations driven by the stronger effect of coercive gradient terms. This feature allows us to construct suitable strict supersolutions from which we conclude Hölder estimates for bounded subsolutions. In many interesting situations, this gives way to a priori estimates for subsolutions. We apply this regularity results to obtain the ergodic asymptotic behavior of the associated evolution problem in the case of superlinear equations. One of the surprising features in our proof is that it avoids the key ingredient which are usually necessary to use the Strong Maximum Principle: linearization based on the Lipschitz regularity of the solution of the ergodic problem. The proof entirely relies on the Hölder regularity.

math.AP↗
arXiv2013

Fractional decay bounds for nonlocal zero order heat equations

In this paper we obtain bounds for the decay rate for solutions to the nonlocal problem $\partial_t u(t,x) = \int_{\R^n} J(x,y)[u(t,y) - u(t,x)] dy$. Here we deal with bounded kernels $J$ but with polynomial tails, that is, we assume a lower bound of the form $J(x,y) \geq c_1|x-y|^{-(n + 2σ)}$, for $|x - y| > c_2$. Our estimates takes the form $\|u(t)\|_{L^q(\R^n)} \leq C t^{-\frac{n}{2σ} (1 - \frac{1}{q})}$ for $t$ large.

math.AP↗
arXiv2013

Existence and Uniqueness for Integro-Differential Equations with Dominating Drift Terms

In this paper we are interested on the well-posedness of Dirichlet problems associated to integro-differential elliptic operators of order $α< 1$ in a bounded smooth domain $Ω$ . The main difficulty arises because of losses of the boundary condition for sub and supersolutions due to the lower diffusive effect of the elliptic operator compared with the drift term. We consider the notion of viscosity solution with generalized boundary conditions, concluding strong comparison principles in $\barΩ$ under rather general assumptions over the drift term. As a consequence, existence and uniqueness of solutions in $C(\barΩ)$ is obtained via Perron's method.

math.AP↗