Equations In Convex Domains Research Articles

Convexity of solutions to elliptic PDE's

This article concerns the convexity or concavity of solutions to special second order elliptic partial differential equations in convex domains. We concentrate our investigation to boundary blow up solutions as well as to solutions of particular singular equations. Following a method due to Korevaar and Kennington, we find a new sufficient condition for proving convexity or concavity. This sufficient condition is useful when the semilinear component of the equation is the sum of two or more terms. For more information see https://ejde.math.txstate.edu/Volumes/2024/51/abstr.html

A high-efficient accurate coupled mesh-free scheme for 2D/3D space-fractional convection-diffusion/Burgers' problems

This paper first develops a high-efficient accurate mesh-free scheme (CSPH-SFCD) for multi-dimensional time-dependent spatial fractional convection-diffusion (SFCD) equations based on the Riemann-Liouville (RL) derivative in convex domain by coupling a corrected smoothed particle hydrodynamics (CSPH) method with a numerical integral of RL. Subsequently, for the first application, the proposed CSPH-SFCD coupled with up-wind (UW) scheme (CSPH-SFCD-UW) is tentatively extended to investigate the nonlinear shock-wave phenomena in 2D space-fractional Burgers' problem (SFBP) with fractional Neumann boundary at high Reynolds number. The proposed mesh-free scheme for time-dependent SFCD/SFBP with fractional boundary is mainly motived by: (a) the SFCD model with RL fractional calculus is discretized by employing a numerical integral formula and a CSPH scheme, and then a conditionally stable CSPH-SFCD method is first derived; (b) the UW scheme is introduced to stabilize the numerical investigation of convection-dominated SFBP; (c) the fractional Neumann boundary (FNB) condition can be easily enforced in the proposed discretized scheme, and the integer-order NB condition is treated by employing virtual particles interpolation technique; (d) the MPI parallel computing technique is adopted to enhance the computational efficiency. Firstly, the numerical convergence and stability of the proposed scheme are discussed and illustrated by solving several multi-dimensional benchmarks. Secondly, the cases of two-sided and left-sided RL fractional derivatives are also discussed and demonstrated in the numerical tests. The advantages of the proposed method are demonstrated by simulating the 2D/3D SFCD equations in convex domains or under non-uniform point distribution. Finally, the 2D two-component SFBP with FNBC are numerically investigated by using the proposed method, and compared with the FDM results. The complex nonlinear non-local properties governed by SFBP are numerically predicted by the proposed scheme. All the numerical results show that the proposed mesh-free method for time-dependent spatial SFCD problems is high-efficient, flexible and accurate. • A high-efficient accurate CSPH-SFCD scheme is first developed to 2D/3D SFCD equations. • The CSPH-SFCD-UW scheme is first proposed to investigate shock-wave behavior in 2D SFBP. • The FNB condition can be easily enforced in the proposed approximated scheme. • The advantages of the present scheme are shown by numerical tests on convex domain.

Boundary differentiability of solutions to elliptic equations in convex domains in the borderline case

In this work, we consider the following elliptic partial differential equations: $$\begin{aligned} \left\{ \begin{aligned} - b_{ij} \, \frac{\partial ^{2} w}{\partial x_{i} \partial x_{j}}&= g \,\,\, \text {in} \,\, \Omega , \\ w&= 0 \,\,\,\text {on} \,\partial \Omega , \end{aligned} \right\} . \end{aligned}$$where the domain \(\Omega \subset {\mathbb {R}}^{n}\) is convex, the matrix \(\big (b_{ij}\big )_{n \times n}\) satisfies the uniform ellipticity conditions. For g in the scaling critical Lorentz space \( L(n,\, 1)(\Omega )\), we establish boundary differentiability of solutions to the above problem. We also prove \(C^{\mathrm {Log-Lip}}\) regularity estimate at a boundary point in the case when \(g \in L^{n}(\Omega )\).

A preconditioner based on sine transform for two-dimensional semi-linear Riesz space fractional diffusion equations in convex domains

In this paper, we develop a fast numerical method for solving the time-dependent Riesz space fractional diffusion equations with a nonlinear source term in the convex domain. An implicit finite difference method is employed to discretize the Riesz space fractional diffusion equations with a penalty term in a rectangular region by the volume-penalization approach. The stability and the convergence of the proposed method are studied. As the coefficient matrix is with the Toeplitz-like structure, the generalized minimum residual method with a preconditioner based on the sine transform is exploited to solve the discretized linear system, where the preconditioner is constructed in view of the combination of two approximate inverse τ matrices, which can be diagonalized by the sine transform. The spectrum of the preconditioned matrix is also investigated. Numerical experiments are carried out to demonstrate the efficiency of the proposed method.

Approximate convexity principles and applications to PDEs in convex domains

We obtain approximate convexity principles for solutions to some classes of nonlinear elliptic partial differential equations in convex domains involving approximately concave nonlinearities. Furthermore, we provide some applications to some meaningful special cases.

A preconditioned fast finite difference scheme for space-fractional diffusion equations in convex domains

A fast finite difference method is developed for solving space-fractional diffusion equations with variable coefficient in convex domains using a volume penalization approach. The resulting coefficient matrix can be written as the discretized matrix from the extended rectangular domain plus a diagonal matrix with jumping entries due to the penalization parameter. An efficient preconditioner is constructed based on the combination of two approximate inverse circulant matrices. The preconditioned BiCGSTAB method, with the proposed preconditioner, is implemented for solving the resulting linear system. Numerical results are carried out to demonstrate the utility of the proposed algorithm.

Optimal Concavity for Newtonian Potentials

In this note I give a short overview about convexity properties of solutions to elliptic equations in convex domains and convex rings and show a result about the optimal concavity of the Newtonian potential of a bounded convex domain in ℝ n , n ≥ 3, namely: if the Newtonian potential of a bounded domain is ”sufficiently concave”, then the domain is necessarily a ball. This result can be considered an unconventional overdetermined problem. This paper is based on a talk given by the author in Bologna at the ”Bruno Pini Mathematical Analysis Seminar”, which in turn was based on the paper P. Salani, A characterization of balls through optimal concavity for potential functions, Proc. AMS 143 (1) (2015), 173-183.

Elliptic equations in convex domains

A short survey of a series of results by the author, partly obtained in collaboration with Yu. Burago.

Lipschitz regularity for a general class of quasilinear elliptic equations in convex domains

We establish a local Lipschitz regularity near the boundary of weak solutions to a general class of homogeneous quasilinear elliptic equations with Neumann boundary condition in bounded convex domains.

A fast finite volume method for conservative space-fractional diffusion equations in convex domains

We develop a fast finite volume method for variable-coefficient, conservative space-fractional diffusion equations in convex domains via a volume-penalization approach. The method has an optimal storage and an almost linear computational complexity. The method retains second-order accuracy without requiring a Richardson extrapolation. Numerical results are presented to show the utility of the method.

Ultraboundedness for parabolic equations in convex domains without boundary conditions

We consider the operator A u = 1 2 Δ u − 〈 D U , D u 〉 , where U is a convex real function defined in a convex open set O ⊂ R N and lim | x | → ∞ U ( x ) = lim x → ∂ O U ( x ) = + ∞ . We prove that the associated Markov semigroup is ultrabounded with respect to the Gibbs measure e − 2 U ( x ) d x .

On a class of elliptic and parabolic equations in convex domains without boundary conditions

We consider the operator $\A u = \frac{1}{2} \Delta u - \langle DU, Du\right$, where $U $ is a convex real function defined in a convex open set $\O \subset \R^N$ and $\lim_{|x|\to \infty} U(x) = \lim_{ x \to \partial \O} U(x)$ $ =$ $ +\infty$. We study the realization of $\A $ in the spaces $C_{b}(\overline{\O})$, $C_{b}(\O)$ and $B_{b}(\O)$, and prove several properties of the associated Markov semigroup. In contrast with the case of bounded coefficients, elliptic equations and parabolic Cauchy problems such as (3) and (4) below are uniquely solvable in reasonable classes of functions, without imposing any boundary condition. We prove that the associated semigroup coincides with the transition semigroup of a stochastic variational inequality on $C_{b}(\overline{\O})$.

Solvability of systems of nonhomogeneous convolution equations in convex domains in $\mathbb{C}$

Solvability of systems of nonhomogeneous convolution equations in convex domains in $\mathbb{C}$

Interior regularity of solutions to a complex Monge-Ampère equation

We give interior estimates for first derivatives of solutions to a type of complex Monge-Ampère equations in convex domains. We also show global estimates for first derivatives of solutions in arbitrary domains. These global estimates are then used to show interior regularity of solutions to the complex Monge-Ampère equations in hyperconvex domains having a bounded exhaustion function which is globally Lipschitz. Finally we give examples of domains which have such an exhaustion function and domains which do not.

Regular growth of systems of functions and systems of non-homogeneous convolution equations in convex domains of the complex plane

In this paper we introduce the notion of regular growth for a system of entire functions of finite order and type. This is a direct and natural generalization of the classical completely regular growth of an entire function. We obtain sufficient and necessary conditions for the solubility of a system of non-homogeneous convolution equations in convex domains of the complex plane. These conditions depend on whether the system of Laplace transforms of the analytic functionals that generate the convolution equations has regular growth. In the case of smooth convex domains, these solubility conditions form a criterion.

Convolution equations in convex domains

Convolution equations in convex domains