اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدRegularity of almost periodic solutions of Poissons equation
146
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
This paper discusses some regularity of almost periodic solutions of the Poissons equation $-Delta u = f$ in $mathbb{R}^n$, where $f$ is an almost periodic function. It has been proved by Sibuya [Almost periodic solutions of Poissons equation. Proc. Amer. Math. Soc., 28:195--198, 1971.] that if $u$ is a bounded continuous function and solves the Poissons equation in the distribution sense, then $u$ is an almost periodic function. In this work, we relax the assumption of the usual boundedness into boundedness in the sense of distribution which we refer to as a bounded generalized function. The set of bounded generalized functions are wider than the set of usual bounded functions. Then, upon assuming that $u$ is a bounded generalized function and solves the Poissons equation in the distribution sense, we prove that this solution is bounded in the usual sense, continuous and almost periodic. Moreover, we show that the first partial derivatives of the solution $partial u/ partial x_i$, $i=1, ldots, n$, are also continuous, bounded, and almost periodic functions. The technique is based on extending a representation formula using Greens function for Poissons equation for solutions in the distribution sense. Some useful properties of distributions are also shown that can be used to study other elliptic problems.
قيم البحث
اقرأ أيضاً
We show that convex viscosity solutions of the Lagrangian mean curvature equation are regular if the Lagrangian phase has Holder continuous second derivatives.
Let $G=(V,E)$ be a finite connected graph, and let $kappa: Vrightarrow mathbb{R}$ be a function such that $int_Vkappa dmu<0$. We consider the following Kazdan-Warner equation on $G$:[Delta u+kappa-K_lambda e^{2u}=0,] where $K_lambda=K+lambda$ and $K:
Vrightarrow mathbb{R}$ is a non-constant function satisfying $max_{xin V}K(x)=0$ and $lambdain mathbb{R}$. By a variational method, we prove that there exists a $lambda^*>0$ such that when $lambdain(-infty,lambda^*]$ the above equation has solutions, and has no solution when $lambdageq lambda^ast$. In particular, it has only one solution if $lambdaleq 0$; at least two distinct solutions if $0<lambda<lambda^*$; at least one solution if $lambda=lambda^ast$. This result complements earlier work of Grigoryan-Lin-Yang cite{GLY16}, and is viewed as a discrete analog of that of Ding-Liu cite{DL95} and Yang-Zhu cite{YZ19} on manifolds.
We prove that Gevrey regularity is propagated by the Boltzmann equation with Maxwellian molecules, with or without angular cut-off. The proof relies on the Wild expansion of the solution to the equation and on the characterization of Gevrey regularity by the Fourier transform.
In this paper we study the Cauchy problem for the Landau Hamiltonian wave equation, with time dependent irregular (distributional) electromagnetic field and similarly irregular velocity. For such equations, we describe the notion of a `very weak solu
tion adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifier of the coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or distributional type solutions under conditions when such solutions also exist.
We investigate the stationary diffusion equation with a coefficient given by a (transformed) Levy random field. Levy random fields are constructed by smoothing Levy noise fields with kernels from the Matern class. We show that Levy noise naturally ex
tends Gaussian white noise within Minlos theory of generalized random fields. Results on the distributional path spaces of Levy noise are derived as well as the amount of smoothing to ensure such distributions become continuous paths. Given this, we derive results on the pathwise existence and measurability of solutions to the random boundary value problem (BVP). For the solutions of the BVP we prove existence of moments (in the $H^1$-norm) under adequate growth conditions on the Levy measure of the noise field. Finally, a kernel expansion of the smoothed Levy noise fields is introduced and convergence in $L^n$ ($ngeq 1$) of the solutions associated with the approximate random coefficients is proven with an explicit rate.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
