Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $μ$. The associated Cameron-Martin space is denoted by $H$. Let $ν=e^{-U}μ$, where $e^{-U}$ is a sufficiently regular weight and $U:X\rightarrow\mathbb{R}$ is a convex and continuous function. In this paper we are interested in the $W^{2,2}$ regularity of the weak solutions of elliptic equations of the type \[λu-L_νu=f,\] where $λ>0$, $f\in L^2(X,ν)$ and $L_ν$ is the self-adjoint operator associated with the quadratic form \[(ψ,φ)\mapsto \int_X\left\langle\nabla_Hψ,\nabla_Hφ\right\rangle_Hdν\qquadψ,φ\in W^{1,2}(X,ν).\]
A unified approach to embedding theorems for Sobolev type spaces of vector-valued functions, defined via their symmetric gradient, is proposed. The Sobolev spaces in question are built upon general rearrangement-invariant norms. Optimal target spaces in the relevant embeddings are determined within the class of all rearrangement-invariant spaces. In particular, all symmetric gradient Sobolev embeddings into rearrangementinvariant target spaces are shown to be equivalent to the corresponding embeddings for the full gradient built upon the same spaces. A sharp condition for embeddings into space…
We develop a functional framework suitable for the treatment of partial differential equations and variational problems on evolving families of Banach spaces. We propose a definition for the weak time derivative that does not rely on the availability of a Hilbertian structure and explore conditions under which spaces of weakly differentiable functions (with values in an evolving Banach space) relate to classical Sobolev--Bochner spaces. An Aubin--Lions compactness result is proved. We analyse concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we…
An elliptic boundary problem acting on generalized Sobolev spaces
本文考虑$\mathbb{R}^n$中有界区域$Ω$上的椭圆边值问题，它作用在$1 < p < \infty$的广义索伯列夫空间$W^{0,χ}_p(Ω)$上.我们注意到，类似的问题$Ω$无论是有界区域在$\mathbb{R}^n$或一个闭流形作用于$W^{0,χ}_2(Ω)$，称为Hörmander空间，已调查的主题由不同的作者。然后在参数椭圆性假设下，建立了边值问题解的存在唯一性的结果。此外，在进一步假设边界条件为零的情况下，我们将建立由边界问题引起的巴拿赫空间算子的谱性质的结果，特别是其特征值的角分布和渐近分布。
Let $X$ be a ball Banach function space on ${\mathbb R}^n$. In this article, under some mild assumptions about both $X$ and the boundedness of the Hardy--Littlewood maximal operator on the associate space of the convexification of $X$, the authors prove that, for any locally integrable function $f$ with $\|\,|\nabla f|\,\|_{X}<\infty$, $$\sup_{λ\in(0,\infty)}λ\left \|\left|\left\{y\in{\mathbb R}^n:\ |f(\cdot)-f(y)| >λ|\cdot-y|^{\frac{n}{q}+1}\right\}\right|^{\frac{1}{q}} \right\|_X\sim \|\,|\nabla f|\,\|_X$$ with the positive equivalence constants independent of $f$, where the index $q\in(0,\i…
By means of Banach fixed point theorem , the uniqueness of Boltzmann Equation generalizaled Solutions in Sobolev spaces in $L^1(\mathbb{R}^+\times \mathbb{R}^n\times \mathbb{R}^n)$, can be proved as well as Boltzmann Equation renormalized solutions uniqueness
A strong inspiration for studying Sobolev type fractional evolution equations comes from the fact that have been verified to be useful tools in the modeling of many physical processes. We introduce a novel technique for solving Sobolev type fractional evolution equations with multi-orders in a Banach space. We propose a new Mittag-Leffler type function which is generated by linear bounded operators and investigate their properties which are productive for checking the candidate solutions for multi-term fractional differential equations. Furthermore, we propose an exact analytical representatio…
This paper investigates the Cauchy problem of the time-space fractional Keller-Segel-Navier- Stokes model, which can describe both memory effect and Lévy process of the system. The local existence and global existence in Lebesgue space are obtained by means of Banach fixed point theorem and Banach implicit function theorem, respectively. In addition, the regularities of local and global mild solutions are improved in fractional homogeneous Sobolev spaces. Furthermore, some properties of mild solutions including mass conservation, decay estimates, stability and self-similarity are established.
We prove the existence of at least three solutions for a weighted $p$-Laplacian operator involving Dirichlet boundary condition in a weighted Sobolev space. The main tool we use here is a three solution theorem in reflexive Banach spaces due to G. Bonanno and B. Ricceri.
This article is a comparative study on an initial-boundary value problem for a class of semilinear pseudo-parabolic equations with the fractional Caputo derivative, also called the fractional Sobolev-Galpern type equations. The purpose of this work is to reveal the influence of the degree of the source nonlinearity on the well-posedness of the solution. By considering four different types of nonlinearities, we derive the global well-posedness of mild solutions to the problem corresponding to the four cases of the nonlinear source terms. For the advection source function case, we apply a nontri…
Higher order weak differentiability and Sobolev spaces between manifolds
定义了从流形$M$到流形$N$的高阶余局部弱可微映射的概念。当$M$和$N$被赋予黎曼度量$p\ge 1$和$k\ge 2$时，这允许我们定义内在高阶齐次索伯列夫空间$\dot{W}^{k,p}(M,N)$。我们表明，这个新的内在的定义是不等价的，一般的定义由等距嵌入的$N$在欧几里得空间;如果流形$M$和$N$是紧凑的，内在的空间是一个更大的空间比一个通过嵌入。我们证明了内在空间$\dot{W}^{k,p}(M,N)$中光滑映射稠密的一个必要条件是$π_{\lfloor k p \rfloor} (N) \simeq \{0\}$ .我们调查的链式规则高阶可微在此设置。
In this paper we provide a complete study of the spectrum of a constant coefficients differential operator on a scale of localized Sobolev spaces, $H^{s}_{loc}(I),$ which are Fréchet spaces. This is quite different from what we find in the literature, where all the relevant results are concerned with spectrum on Banach spaces. Our aim is to understand the behavior of all the three types of spectrum (point, residual and continuous) and the relation between them and those of the dual operator. The main result we present shows that there is no complex number in the resolvent set of such operators…
In this paper, we prove global-in-time $\dot{\mathrm{H}}^{α,q}$-maximal regularity for a class of injective, but not invertible, sectorial operators on a UMD Banach space X , provided $q\in(1,+\infty) , $α\in(-1+1/q,1/q)$. We also prove the corresponding trace estimate, so that the solution to the canonical abstract Cauchy problem is continuous with values in a not necessarily complete trace space.In order to put our result in perspective, we also provide a short review on L q-maximal regularity which includes some recent advances such as the revisited homogeneous operator and interpolation th…
Lojasiewicz-Simon gradient inequalities for analytic and Morse-Bott functions on Banach spaces
我们证明了巴拿赫空间上解析泛函的Lojasiewicz-Simon梯度不等式的几个抽象版本，推广了这个不等式的以前的抽象版本，削弱了他们的假设，特别是著名的无穷维版本的梯度不等式，由于Lojasiewicz证明了西蒙（1983）。我们还证明了当泛函为Morse-Bott时，Lojasiewicz-Simon梯度不等式的最优指数是得到的，改进了Chill（2003，2006），Haraux and Jendoubi（2007）和Simon（1996）的类似结果。在我们的文章arXiv：1903.01953中，我们应用抽象的Lojasiewicz-Simon梯度不等式证明了调和映射能量泛函的Lojasiewicz-Simon梯度不等式，该能量泛函使用索伯列夫空间，对闭黎曼流形之间的映射施加最小正则性要求。调和映射能量泛函的这些不等式推广了Kwon（2002）、Liu and Yang（2010）、Simon（1983，1985）和Topping（1997）的不等式。在我们的专著arXiv：1510.03815中，我们证明了Lojasiewicz-Simon梯度不等式耦合杨-米尔斯能量函数使用索伯列夫空间施加最小的正则性要求对连接和部分。这些不等式推广了第一作者（arXiv：1409.1525中的定理23.1和23.17）关于任意维基流形和拉德（1992）关于二维和三维基流形的纯杨-米尔…
We consider the spaces $L^p(X,ν;V)$, where $X$ is a separable Banach space, $μ$ is a centred non-degenerate Gaussian measure, $ν:=Ke^{-U}μ$ with normalizing factor $K$ and $V$ is a separable Hilbert space. In this paper we prove a vector-valued Poincaré inequality for functions $F\in W^{1,p}(X,ν;V)$, which allows us to show that for every $p\in(1,+\infty)$ and every $k\in\mathbb N$ the norm in $W^{k,p}(X,ν)$ is equivalent to the graph norm of $D_H^k$ (the $k$-th Malliavin derivative) in $L^p(X,ν)$. To conclude, we show exponential decay estimates for $(T^V(t))_{t\geq0}$ as $t\rightarrow+\infty…
