M. A. Ragusa, A. Scapellato
In this paper, new classes of functions are defined. These spaces generalize Morrey spaces and give a refinement of Lebesgue spaces. Some embeddings between these new classes are also proved. Finally, the authors apply these classes of functions to obtain regularity results for solutions of partial differential equations of parabolic type.
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The Navier-Stokes equations in mixed-norm time-space parabolic Morrey spaces
本文研究外力属于 Krylov 混合范数时空抛物 Morrey 空间时的 Navier-Stokes 方程。
On solvability of parabolic equations with singular coefficients in odd mixed-norm Morrey-Sobolev spaces
本文研究了在奇混合范数Morrey- Sobolev空间中，具有奇异系数的抛物方程的可解性问题，通过引入混合范数Morrey类和BMO条件，证明了存在性和唯一性定理。
Besov mixed-Morrey spaces: On an Application to the Navier-Stokes Equations
本文引入Besov混合-Morrey空间与Fourier-Besov混合-Morrey空间，并利用Bony仿积方法证明d维不可压缩Navier-Stokes方程在小初值下的全局适定性，给出具有各向异性的新初值类。
On parabolic equations in Morrey spaces with VMO $a$ and Morrey $b,c$
本文研究了全空间中具有 VMO $a$ 和 Morrey $b,c$ 系数的二阶抛物方程，证明了其解在混合范数 Morrey 空间中的存在性与唯一性。
We present several results of embedding type for parabolic Morrey spaces with or without mixed norms. Some other interpolation results for parabolic Morrey spaces are also given.
Partial regularity for variational integrals with Morrey-Hölder zero-order terms, and the limit exponent in Massari's regularity theorem
本文研究了变分积分中带有Morrey-Hölder零阶项的部分正则性，验证了Massari正则性定理中极限指数的最优正则性结果。
$C^{1,α}$-regularity for Mixed Local and Nonlocal Degenerate Elliptic Equations in the Heisenberg Group
本文在Heisenberg群上研究混合局部与非局部退化椭圆方程弱解的$C^{1,α}$正则性，通过Morrey型迭代方案和Hölder连续性证明了弱解的正则性。
We generalize here the celebrated Partial Regularity Theory of Caffarelli, Kohn and Nirenberg to the MHD equations in the framework of parabolic Morrey spaces. This type of parabolic generalization using Morrey spaces appears to be crucial when studying the role of the pressure in the regularity theory for the classical Navier-Stokes equations as well as for the MHD equations.
In this paper we prove Fefferman's inequalities associated to potentials belonging to a generalized Morrey space $ L^{p,φ} $ or a Stummel class $ \tilde{S}_{α,p} $. Our results generalize and extend Fefferman's inequalities obtained in \cite{CRR,CF,F,Z1}. We also show that the logarithmic of non-negative weak solution of second order elliptic partial differential equation, where its potentials are assumed in generalized Morrey spaces and Stummel classes, belongs to the bounded mean oscillation class. As a consequence, this elliptic partial differential equation has the strong unique continuati…
We presents the study the separability properties for differential-operator equations in Morrey spaces. We prove that the corresponding differential operator is a generator of analytic semigroup in vector-valued Morrey spaces. Moreover, maximal regularity properties of corresponding parabolic equation ıs obtained. In applications, the maximal regularity properties of Wentzell-Robin type problem for elliptic equations and mixed value problem for degenerate parabolic equations in Morrey spaces are derived.
Barker recently proved new weak-strong uniqueness results for the Navier-Stokes equations based on a criterion involving Besov spaces and a proof through interpolation between Besov-H{ö}lder spaces and L 2. We improve slightly his results by considering Besov-Morrey spaces and interpolation between Besov-Morrey spaces and L 2 uloc. Let u 0 a divergence-free vector field on R 3. We shall consider weak solutions to the Cauchy initial value problem for the Navier-Stokes equations which satisfy energy estimates. The differential Navier-Stokes equations read as $\partial$ t u + u. $\nabla$ u = $Δ$ …
On parabolic equations in Sobolev spaces with lower-order coefficients from Morrey spaces
该研究针对低阶系数取自Morrey空间的抛物方程，证明了在有界$C^{1,1}$柱体中，当右端项属于$L_p$时，方程在Sobolev空间中具有可解性。
Essentials of Real Analysis and Morrey-Sobolev spaces for second-order elliptic and parabolic PDEs with singular first-order coefficients
本文简要介绍了Hardy-Littlewood极大函数定理等实分析结果在具奇异一阶系数的二阶椭圆与抛物型PDE的Sobolev及Morrey-Sobolev空间理论中的应用，并在最后一节给出了新结果。
Morrey-Lorentz estimates for Hodge-type systems
本文证明了具有正则各向异性系数的Hodge型微分形式系统弱解在Morrey-Lorentz空间中的到边界正则性估计，并探讨了其在Hodge分解及相关系统中的应用。
Derivative Embeddings in Vector-Valued Morrey Spaces
本文为具有Morrey时间控制的Banach值演化类建立中间导数嵌入定理，通过有理频率分裂和K-泛函证明无需UMD等假设，并给出最优指标、迹公式及紧性结果。
The aim of this paper is to present the global estimate for gradient of renormalized solutions to the following quasilinear elliptic problem: \begin{align*} \begin{cases} -div(A(x,\nabla u)) &= μ\quad \text{in} \ \ Ω, \\ u &=0 \quad \text{on} \ \ \partial Ω, \end{cases} \end{align*} in Lorentz-Morrey spaces, where $Ω\subset \mathbb{R}^n$ ($n \ge 2$), $μ$ is a finite Radon measure, $A$ is a monotone Carathéodory vector valued function defined on $W^{1,p}_0(Ω)$ and the $p$-capacity uniform thickness condition is imposed on the complement of our domain $Ω$. It is remarkable that the local gradien…
We study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point $ζ\in\partialΩ\cup\{\infty\}$ of the quasilinear elliptic equations $$-\text{div}(|\nabla u|_A^{p-2}A\nabla u)+V|u|^{p-2}u =0\quad\text{in } Ω\setminus\{ζ\},$$ where $Ω$ is a domain in $\mathbb{R}^d$ ($d\geq 2$), and $A=(a_{ij})\in L_{\rm loc}^{\infty}(Ω;\mathbb{R}^{d\times d})$ is a symmetric and locally uniformly positive definite matrix. The potential $V$ lies in a certain local Morrey space (depending on $p$) and has a Fuchsian-type isolated singularity at $ζ$.
Local regularity results are obtained for the MHD equations using as global framework the setting of parabolic Morrey spaces. Indeed, by assuming some local boundedness assumptions (in the sense of parabolic Morrey spaces) for weak solutions of the MHD equations it is possible to obtain a gain of regularity for such solutions in the general setting of the Serrin regularity theory. This is the first step of a wider program that aims to study both local and partial regularity theories for the MHD equations.
In this paper we strengthen to Morrey-Lorentz spaces the famous trace principle introduced by Adams. More precisely, we show that Riesz potential $I_α$ is continuous \begin{equation} \Vert I_αf\Vert_{\mathcal{M}_{q, \infty}^{λ_{\ast}}(dμ)}\lesssim \Arrowvertμ\Arrowvert_β^{{1}/{q}}\,\Vert f\Vert_{\mathcal{M}_{p, \infty}^λ(dν)}\nonumber\\[0.02in] \end{equation} if and only if the Radon measure $dμ$ supported in $Ω\subset \mathbb{R}^n$ is controlled by $$\Arrowvertμ\Arrowvert_β=\sup_{x\in\mathbb{R}^n,\,r>0}r^{-β}μ(B(x,r))<\infty$$ provided that $1<p<q<\infty$ satisfies $n-αp<β\leq n,\; α=\frac{n}…
The semilinear heat inequality with Morrey initial data on Riemannian manifolds
研究有界几何黎曼流形上具有临界 Morrey 范数小初始数据的半线性热不等式，通过假设初始数据的 Morrey 范数和解的 L^s 范数有界，得到解的 L^∞ 估计，并改进了初始时刻附近的估计，结果可应用于高维几何流。
In this paper, we are concerned with the partial regularity of the suitable weak solutions to the fractional MHD equations in $\mathbb{R}^{n}$ for $n=2,\,3$. In comparison with the work of the 3D fractional Navier-Stokes equations obtained by Tang and Yu in [24, Commun. Math. Phys. 334: 1455--1482, 2015], our results include their endpoint case $α=3/4$ and the external force belongs to more general parabolic Morrey space. Moreover, we prove some interior regularity criteria just via the scaled mixed norm of the velocity for the suitable weak solutions to the fractional MHD equations.
Weighted Orlicz-Sobolev and variable exponent Morrey regularity for fully nonlinear parabolic PDEs with oblique boundary conditions and applications
本文针对带斜边界条件的完全非线性抛物方程粘性解，建立全局加权Orlicz-Sobolev与变指数Morrey-Sobolev估计，给出稠密性、障碍问题正则性等相关应用。
This paper continues the development of regularity results for quasilinear measure data problems \begin{align*} \begin{cases} -\mathrm{div}(A(x,\nabla u)) &= μ\quad \text{in} \ \ Ω, \\ \quad \quad \qquad u &=0 \quad \text{on} \ \ \partial Ω, \end{cases} \end{align*} in Lorentz and Lorentz-Morrey spaces, where $Ω\subset \mathbb{R}^n$ ($n \ge 2$), $μ$ is a finite Radon measure on $Ω$, and $A$ is a monotone Carathéodory vector valued operator acting between $W^{1,p}_0(Ω)$ and its dual $W^{-1,p'}(Ω)$. It emphasizes that this paper studies the `very singular' case $1<p \le \frac{3n-2}{2n-1}$ and th…
In this note we give a proof of the Sobolev and Morrey embedding theorems based on the representation of functions in terms of the fundamental solution of suitable partial differential operators. We also prove the compactness of the Sobolev embedding. We first describe this method in the classical setting, where the fundamental solution of the Laplace equation is used, to recover the classical Sobolev and Morrey theorems. We next consider degenerate Kolmogorov equations. In this case, the fundamental solution is invariant with respect to a non-Euclidean translation group and the usual convolut…
The targets of this article are threefold. The first one is to give a survey on the recent developments of function spaces with mixed norms, including mixed Lebesgue spaces, iterated weak Lebesgue spaces, weak mixed-norm Lebesgue spaces and mixed Morrey spaces as well as anisotropic mixed-norm Hardy spaces. The second one is to provide a detailed proof for a useful inequality about mixed Lebesgue norms and the Hardy--Littlewood maximal operator and also to improve some known results on the maximal function characterizations of anisotropic mixed-norm Hardy spaces and the boundedness of Calderón…
The present work constitutes a first step towards establishing a systematic framework for treating variational problems that depend on a given input function through a mixture of its derivatives of different orders in different directions. For a fixed vector $\mathbf{a} := (a_1, \ldots, a_N) \in \mathbb{N}^N$ and $u \colon \mathbb{R}^N \supset Ω\to \mathbb{R}^n$ we denote by $\nabla_{\mathbf{a}} u := (\partial^α u)_{\langle α, \mathbf{a}^{-1} \rangle = 1}$ the matrix whose $i$-th row is composed of derivatives $\partial^αu^i$ of the $i$-th component of the map $u$, and where the multi-indices …
Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients on $\mathbb{R}^n$ and $X$ a ball quasi-Banach function space on $\mathbb{R}^n$ satisfying some mild assumptions. Denote by $H_{X,\, L}(\mathbb{R}^n)$ the Hardy space, associated with both $L$ and $X$, which is defined via the Lusin area function related to the semigroup generated by $L$. In this article, the authors establish both the maximal function and the Riesz transform characterizations of $H_{X,\, L}(\mathbb{R}^n)$. The results obtained in this article have a wide range o…
Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regularity and uniqueness of a weak solution is obtained for the 2D MHD equations with only magnetic diffusion.
This paper focuses on the 3D incompressible magnetohydrodynamic (MHD) equations with mixed partial dissipation and magnetic diffusion. Our main result assesses the global stability of perturbations near the steady solution given by a background magnetic field. The stability problem on the MHD equations with partial or no dissipation has attracted considerable interests recently and there are substantial developments. The new stability result presented here is among the very few stability conclusions currently available for ideal or partially dissipated MHD equations. As a special consequence o…
Second order parabolic equations in Sobolev spaces with mixed norms are studied. The leading coefficients (except $a^{11}$) are measurable in both time and one spatial variable, and VMO in the other spatial variables. The coefficient $a^{11}$ is measurable in time and VMO in the spatial variables. The unique solvability of equations in the whole space is applied to solving Dirichlet and oblique derivative problems for parabolic equations defined in a half-space.
Mixed-dimensional partial differential equations arise in several physical applications, wherein parts of the domain have extreme aspect ratios. In this case, it is often appealing to model these features as lower-dimensional manifolds embedded into the full domain. Examples are fractured and composite materials, but also wells (in geological applications), plant roots, or arteries and veins. In this manuscript, we survey the structure of mixed-dimensional PDEs in the context where the sub-manifolds are a single dimension lower than the full domain, including the important aspect of intersecti…
A regularity theory for second-order parabolic partial differential equations in weighted mixed norm Sobolev-Zygmund spaces
本文在加权混合范数 Sobolev-Zygmund 空间中建立了抛物型偏微分方程的最优正则性理论，将经典 Schauder 估计推广到时间仅可测系数和整数阶正则性的临界情形，并通过尖锐迹定理处理非零初值。
Global Hölder solvability of second order elliptic equations with locally integrable lower-order coefficients
本文证明了一类具有低阶项、系数受边界距离负幂控制的椭圆方程全局Hölder连续解的存在性，改进了经典定理，并采用Ancona策略结合新Morrey型空间进行证明。
This note studies local integral gradient bounds for distributional solutions of a large class of partial differential inequalities with diffusion in divergence form and power-like first-order terms. The applications of these estimates are two-fold. First, we show the (sharp) global Hölder regularity of distributional semi-solutions to this class of diffusive PDEs with first-order terms having supernatural growth and right-hand side in a suitable Morrey class posed on a bounded and regular open set $Ω$. Second, we provide a new proof of entire Liouville properties for inequalities with superli…
In this paper, we discuss with the global well-posedness of 2D anisotropic nonlinear Boussinesq equations with any two positive viscosities and one positive thermal diffusivity. More precisely, for three kinds of viscous combinations, we obtain the global well-posedness without any assumption on the solution. For other three difficult cases, under the minimal regularity assumption, we also derive the unique global solution. To the authors' knowledge, our result is new even for the simplified model.
In this paper, we study the nonexistence of positive solutions for the following two mixed boundary value problems. The first problem is the mixed nonlinear-Neumann boundary value problem $$ \left\{ \begin{array}{ll} \displaystyle -Δu=f(u) &{\rm in}\quad \R, \\ \displaystyle \\ \frac{\partial u}{\partial ν}=g(u) &{\rm on}\quad Γ_1,\\ \displaystyle \\ \frac{\partial u}{\partial ν}=0 &{\rm on}\quad Γ_0 \end{array} \right. $$ and the second is the nonlinear-Dirichlet boundary value problem $$ \left\{ \begin{array}{ll} \displaystyle -Δu=f(u) &{\rm in}\quad \R, \\ \displaystyle \\ \frac{\partial u}…
The paper is devoted to the study of positive solutions of a second-order linear elliptic equation in divergence form in a domain $D\subseteq \mathbb{R}^n$ that satisfy an oblique boundary condition on a portion of $\partial D$. First, we study the degenerate mixed boundary value problem $$ \begin{cases} Pu=f & \text{in } D, \\ Bu = 0 & \text{on } \partial D_{\mathrm{Rob}}, \\ u=0& \text{on } \partial D_{\mathrm{Dir}}, \end{cases} $$ where $D$ is a bounded Lipschitz domain, $\partial D_{\mathrm{Rob}}$ is a relatively open portion of $\partial D$, $\partial D_{\mathrm{Dir}}$ is a closed set of …
In this work we study the existence of solutions to the critical Brezis-Nirenberg problem when one deals with the spectral fractional Laplace operator and mixed Dirichlet-Neumann boundary conditions, i.e., $$ \left\{\begin{array}{rcl} (-Δ)^su & = & λu+u^{2_s^*-1},\quad u>0\quad\mbox{in}\quad Ω,\\ u & = & 0\quad\mbox{on}\quad Σ_{\mathcal{D}},\\ \displaystyle\frac{\partial u}{\partial ν} & = & 0\quad\mbox{on}\quad Σ_{\mathcal{N}}, \end{array}\right. $$ where $Ω\subset\mathbb{R}^N$ is a regular bounded domain, $\frac{1}{2}<s<1$, $2_s^*$ is the critical fractional Sobolev exponent, $0\leλ\in \math…
Let $Ω$ be a smooth bounded domain of $\mathbb{R}^{N+1}$ of boundary $\partial Ω= Γ_1 \cup Γ_2$ and such that $\partial Ω\cap Γ_2$ is a neighborhood of $0$, $h \in \mathcal{C}^0(\partial Ω\cap Γ_2) $ and $s \in [0,1)$. We propose to study existence of positive solutions to the following Hardy-Sobolev trace problem with mixed boundaries conditions \begin{align} \begin{cases} Δu= 0& \qquad \textrm{ in } Ω\\\ u=0 & \qquad \textrm{ on } Γ_1 \\\ \frac{\partial u}{\partial ν}=h(x) u + \frac{u^{q(s)-1}}{d(x)^{s}} & \qquad \textrm{ on } Γ_2, \end{cases} \end{align} where $q(s):=\frac{2(N-s)}{N-1}$ is …
In this paper, we establish a new improved Sobolev inequality based on a weighted Morrey space. To be precise, there exists $C=C(n,m,s,α)>0$ such that for any $u,v \in {\dot{H}}^s(\mathbb{R}^{n})$ and for any $θ\in (\barθ,1)$, it holds that \begin{equation} \label{eq0.3} \Big( \int_{ \mathbb{R}^{n} } \frac{ |(uv)(y)|^{\frac{2^*_{s}(α)}{2} } } { |y'|^α } dy \Big)^{ \frac{1}{ 2^*_{s} (α) }} \leq C ||u||_{\dot{H}^s(\mathbb{R}^{n})}^{\fracθ{2}} ||v||_{\dot{H}^s(\mathbb{R}^{n})}^{\fracθ{2}} ||(uv)||^{\frac{1-θ}{2}}_{ L^{1,n-2s+r}(\mathbb{R}^{n},|y'|^{-r}) }, \end{equation} where $s \!\in\! (0,1)$, …
In this article we consider the simultaneous recovery of bulk and boundary potentials in (degenerate) elliptic equations modelling (degenerate) conducting media with inaccessible boundaries. This connects local and nonlocal Calderón type problems. We prove two main results on these type of problems: On the one hand, we derive simultaneous bulk and boundary Runge approximation results. Building on these, we deduce uniqueness for localized bulk and boundary potentials. On the other hand, we construct a family of CGO solutions associated with the corresponding equations. These allow us to deduce …
In this paper we are interested on solvability of the problem \begin{align*} \begin{cases} -Δu=0 & \text{in} \;\;\;\mathbb{R}^{n+1}_{+}\;\;\;\;\;\;\;\;\;\\ \;\;\displaystyle{\frac{\partial u}{\partial ν}} = V(x)u+b \vert u\vert^{ρ-1}u+f \; & \text{on} \;\;\partial\mathbb{R}^{n+1}_+\;\;\;\;\;\;\;\;\,\, \end{cases} \end{align*} %Laplace equation in the upper half-space with nonlinear Neumann boundary with high singular data $f$ and potential $V$ on boundary $\partial\mathbb{R}^{n+1}_+$ of half-space $ \mathbb{R}^{n+1}_{+}=\{(x,t)\in\mathbb{R}^{n+1}\,\vert\, t>0\}$ for $n\geq 2$. More precisely, …
On the stability and exponential decay of the 3D MHD system with mixed partial dissipation near a equilibrium state
本文研究带混合部分耗散的三维MHD方程在背景磁场附近的稳定性，通过分解解的水平模并结合强庞加莱型不等式证明全局稳定性，还得出非零模在H²空间指数衰减、解收敛于水平零模的大时间行为结论。
We define a homogeneous parabolic De Giorgi classes of order 2 which suits a mixed type class of evolution equations whose simplest example is $μ(x) \frac{\partial u}{\partial t} - Δu = 0$ where $μ$ can be positive, null and negative, so in particular elliptic-parabolic and forward-backward parabolic equations are included. For functions belonging to this class we prove local boundedness and show a Harnack inequality which, as by-products, gives Hölder-continuity, in particular in the interface $I$ where $μ$ change sign, and a maximum principle.
In this paper, we consider discrete $p$-Laplacian parabolic equations with $q$-reaction term under the mixed boundary condition and the initial condition as follows: \begin{equation*} \begin{cases} u_{t}\left(x,t\right) = Δ_{p,ω} u\left(x,t\right) +λ\left\vert u\left(x,t\right) \right\vert^{q-1} u\left(x,t\right), &\left(x,t\right) \in S \times \left(0,\infty\right), \\ μ(z)\frac{\partial u}{\partial_{p} n}(z)+σ(z)\vert u(z)\vert^{p-2}u(z)=0, &\left(x,t\right) \in \partial S \times \left[0,\infty\right), \\ u\left(x,0\right) = u_{0}(x) \geq 0, &x \in \overline{S}. \end{cases} \end{equation*} w…
Stability and inviscid limit of the 3D anisotropic MHD system near a background magnetic field with mixed fractional partial dissipation
本文研究三维各向异性 MHD 系统在背景磁场附近的全局稳定性，并证明垂直粘性消失时在 $H^1$ 范数下的收敛极限。
We study the existence of a solution to the mixed boundary value problem for Helmholtz and Poisson type equations in a bounded Lipschitz domain $Ω\subset\mathbb{R}^N$ and in $\mathbb{R}^N\setminusΩ$ for $N\geq3$. The boundary $\partialΩ$ of $Ω$ is the decomposition of $Γ_1,Γ_2\subset\partialΩ$ such that $\partialΩ=Γ=\overlineΓ_1\cupΓ_2=Γ_1\cup\overlineΓ_2$ and $Γ_1\capΓ_2=\emptyset$. We have shown that if the Neumann data $f_2\in H^{-\frac{1}{2}}(Γ_2)$ and the Dirichlet data $f_1\in H^{\frac{1}{2}}(Γ_1)$ then the Helmholtz problem with mixed boundary data admits a unique solution. We have also…
Spectrum properties of mixed operators under the mixed boundary conditions
本文研究了在混合边界条件下，经典拉普拉斯算子与分数拉普拉斯算子叠加的谱性质。
In this work we propose a kinetic formulation for evolutionary game theory for zero sum games when the agents use mixed strategies. We start with a simple adaptive rule, where after an encounter each agent increases the probability of play the successful pure strategy used in the match. We derive the Boltzmann equation which describes the macroscopic effects of this microscopical rule, and we obtain a first order, nonlocal, partial differential equation as the limit when the probability change goes to zero. We study the relationship between this equation and the well known replicator equations…
We consider a (generally, non-coercive) mixed boundary value problem in a bounded domain $D$ of ${\mathbb R}^n$ for a second order parameter-dependent elliptic differential operator $A (x,\partial, λ)$ with complex-valued essentially bounded measured coefficients and complex parameter $λ$. The differential operator is assumed to be of divergent form in $D$, the boundary operator $B (x,\partial)$ is of Robin type with possible pseudo-differential components on $\partial D$. The boundary of $D$ is assumed to be a Lipschitz surface. Under these assumptions the pair $(A (x,\partial, λ),B)$ induces…
This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain $Ω\subset \reals^n$, $n\geq2$, with boundary that is decomposed as $\partialΩ=D\cup N$, $D$ and $N$ disjoint. We let $Λ$ denote the boundary of $D$ (relative to $\partialΩ$) and impose conditions on the dimension and shape of $Λ$ and the sets $N$ and $D$. Under these geometric criteria, we show that there exists $p_0>1$ depending on the domain $Ω$ such that for $p$ in the interval $(1,p_0)$, the mixed problem with Neumann data in the space $L^p(N)$ and Dirichlet data in the Sobolev spa…
A sharp eigenvalue theorem for mixed elliptic problems under mixed boundary conditions
本文针对混合边界条件下含经典与分数阶拉普拉斯算子的混合椭圆问题，建立其正弱解存在性的特征值刻画定理，拓展了经典椭圆框架的相关结果。
We obtain weighted $L^2$ Strichartz estimates for Schrödinger equations $i\partial_tu+(-Δ)^{a/2}u=F(x,t)$, $u(x,0)=f(x)$, of general orders $a>1$ with radial data $f,F$ with respect to the spatial variable $x$, whenever the weight is in a Morrey-Campanato type class. This is done by making use of a useful property of maximal functions of the weights together with frequency-localized estimates which follow from using bilinear interpolation and some estimates of Bessel functions. As consequences, we give an affirmative answer to a question posed in \cite{BBCRV} concerning weighted homogeneous St…
In this work we study the existence of solutions to the following critical fractional problem with concave-convex nonlinearities, \begin{equation*} \left \{ \begin{array}{l} (-Δ)^su=λu^q+u^{2_s^*-1},\ u>0\quad\text{in }Ω,\\[3pt] \mkern+51mu u=0\quad\text{on } Σ_{\mathcal{D}}\\ \mkern+36mu \displaystyle \frac{\partial u}{\partial ν}=0\quad\text{on } Σ_{\mathcal{N}} \end{array} \right. \end{equation*} where $Ω\subset\mathbb{R}^N$ is a smooth bounded domain, $\frac{1}{2}<s<1$, $0<q<2_s^*-1$, $q\neq 1$, being $2_s^*=\frac{2N}{N-2s}$ the critical fractional Sobolev exponent, $λ>0$, $ν$ is the outwa…
Nonlocal critical exponent singular problems under mixed Dirichlet-Neumann boundary conditions
本文研究带混合Dirichlet-Neumann边界条件的分数阶拉普拉斯奇异临界问题，利用Nehari流形技巧证明合适参数范围内至少存在两个能量异号的解。
The non-local problem is considered for the partial differential equation of mixed-type with Bessel operator and fractional order. An explicit solution is represented by Fourier-Bessel series in the given domain. It is established the connection between the given data and the unique solvability of the problem.
Mixed quantization and partial hyperbolicity
本文结合混合量子化解析技术与部分双曲系统的稳定遍历性结果，为自旋 Hamilton 量建立了稳定的量子遍历性。
Let $D$ be a non-empty open subset of $\R^m,\,m\ge 2$, with boundary $\partial D$, with finite Lebesgue measure $|D|$, and which satisfies a parabolic Harnack principle. Let $K$ be a compact, non-polar subset of $D$. We obtain the leading asymptotic behaviour as $\varepsilon\downarrow 0$ of the $L^{\infty}$ norm of the torsion function with a Neumann boundary condition on $\partial D$, and a Dirichlet boundary condition on $\partial (\varepsilon K)$, in terms of the first eigenvalue of the Laplacian with corresponding boundary conditions. These estimates quantify those of Burdzy, Chen and Mars…
Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation, the harmonic oscillator evolution equation $$\partial_tu=Δu-|x|^2u+f,$$ and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-Δ)^su=f$ and $(\partial_t-Δ+|x|^2)^su=f$, for $0<s<1$.
We prove that given any positive integer $k$, for each open set $Ω$ and any closed subset $D$ of its closure such that $Ω$ is locally an (epsilon,delta)-domain near points in the boundary of $Ω$ not contained in $D$ there exists a linear and bounded extension operator $E$ mapping, for each $p\in[1,\infty]$, the space $W^{k,p}_D(Ω)$ into $W^{k,p}_D({\mathbb{R}}^n)$. Here, with $O$ denoting either $Ω$ or the entire ambient, the space $W^{k,p}_D(O)$ is defined as the completion in the classical Sobolev space $W^{k,p}(O)$ of compactly supported smooth functions whose supports are disjoint from $D$…
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