Nikolai Ubostad
We study the boundary regularity for the normalised $\infty$-heat equation $u_t = Δ_{\infty}^Nu$ in arbitrary domains. Perron's Method is used for constructing solutions. We characterize regular boundary points with barrier functions, and prove an Exterior Ball result. A Petrovsky-like criterion is established.
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We show that smallness of $\dot B^{-1}_{\infty,\infty}$ norm of solution to $d$-dimensional ($d\ge 3$) incompressible Navier-Stokes prevents blowups.
In this paper, we prove the global $C^\infty$ regularity of the Oleinik's solution for the steady Prandtl equation with favorable pressure.
We continue our study of the free boundary regularity in the thin one-phase problem and show that $C^{2,α}$ free boundaries are smooth.
In this paper, we investigate regularity criterion for the solution of the nematic liquid crystal flows in dimension three and two. We prove the solution $(u,d)$ is smooth up to time $T$ provided that there exists a positive constant $\varepsilon_{0}>0$ such that (i) for n=3, |(u,\nabla d)|_{L^{\infty}(0,T;\dot{B}^{-1}_{\infty,\infty})}\leq \varepsilon_{0}, and (ii) for $n=2$, |\nabla d|_{L^{\infty}(0,T;\dot{B}^{-1}_{\infty,\infty})}\leq \varepsilon_{0}.
We prove that if $u$ is a suitable weak solution to the three dimensional Navier-Stokes equations from the space $L_{\infty}(0,T;\dot{B}_{\infty,\infty}^{-1})$, then all scaled energy quantities of $u$ are bounded. As a consequence, it is shown that any axially symmetric suitable weak solution $u$, belonging to $L_{\infty}(0,T;\dot{B}_{\infty,\infty}^{-1})$, is smooth.
On the regularity of weak solutions of the 3D Navier-Stokes equations in $B^{-1}_{\infty,\infty}$
证明了如果三维斯托克斯方程的Leray-Hopf解$u$属于$C((0,T]; B^{-1}_{\infty,\infty})$或其在$B^{-1}_{\infty,\infty}$ -范数上的跳跃不超过粘性的常数倍，则$u$在$(0,T]$上是正则的.我们的方法使用频率的非线性项的局部估计，并产生一个经典的Ladyzhenskaya-Prodi-Serrin准则的扩展。
We give a new proof for the $C^\infty$ regularity of $C^2$ smooth conformal mappings of the sub-Riemannian Heisenberg group. Our proof avoids any use of nonlinear potential theory and relies only on hypoellipticity of Hörmander operators and quasiconformal flows. This approach is inspired by prior work of Sarvas and Liu.
Higher regularity estimates for solutions to $\infty$-Laplacian-type models
本文研究了非齐次$\infty$-拉普拉斯方程在内部临界点的高阶正则性，获得了优于有界强迫项模型的平滑性估计，并与障碍及死核型问题的经典结果建立了联系。
In the seminal paper [Arch. Ration. Mech. Anal. 176 (2005), 351--361], Savin proved the $C^1$-regularity of planar $\infty$-harmonic functions $u$. Here we give a new understanding of it from a capacity viewpoint and drop several high technique arguments therein. Our argument is essentially based on a topological lemma of Savin, a flat estimate by Evans and Smart, % \cite{es11a}, $W^{1,2}_{loc}$-regularity of $|Du|$ and Crandall's flow for infinity harmonic functions.
Regularity from $p$-harmonic potentials to $\infty$-harmonic potentials in convex rings
本文在任意凸环域中研究从p-调和势到∞-调和势的正则性，证明了∞-调和势的内部C^1正则性及部分Sobolev正则性。
In my previous paper I have contrived a Ginzburg-Landau heat flow with a time-dependent parameter and by using it, I constructed a harmonic heat flow into spheres with a monotonical inequality and a reverse Poincaré inequality. This paper establishes these two energy inequalities near the boundary and then by making the best of them, we discuss a partial boundary regularity. In addition to it, we demonstrate a whole domain's regularity under "the one-sided condition." This has been proposed by S.Hildebrandt and K.-O.Widman.
We establish a regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity and vacuum in a bounded domain.
We prove that on an asymptotically Euclidean boundary groupoid, the heat kernel of the Laplacian is a smooth groupoid pseudo-differential operator.
Regularity and explicit $L^\infty$ estimates for a class of nonlinear elliptic systems
该研究针对边界具有临界或次临界增长的非线性椭圆耦合系统，利用De Giorgi-Nash-Moser迭代格式及Gagliardo--Nirenberg插值不等式，确立了弱解的$L^\infty$正则性并给出显式估计。
Optimal conditions for $L^\infty$-regularity and a priori estimates for elliptic systems, I: two components
本文对具有两个未知函数的椭圆型方程组提出了一种新的bootstrap方法。结合$L^p$ -估计，给出了三类著名弱解：$H_0^1$ -解、$L^1$ -解和$L^1_δ$ -解的最优$L^\infty$ -正则性条件.由于$L^p_δ(Ω)$中的线性理论，它也产生了$L^1_δ$ -解的先验估计的最优条件。基于先验估计，我们改进了几类椭圆型方程组的存在性定理。
We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set $Ω\subset\mathbb{R}^N$. Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.
Let $(u, π)$ with $u=(u_1,u_2,u_3)$ be a suitable weak solution of the three dimensional Navier-Stokes equations in $\mathbb{R}^3\times [0, T]$. Denote by $\dot{\mathcal{B}}^{-1}_{\infty,\infty}$ the closure of $C_0^\infty$ in $\dot{B}^{-1}_{\infty,\infty}$. We prove that if $u\in L^\infty(0, T; \dot{B}^{-1}_{\infty,\infty})$, $u(x, T)\in \dot{\mathcal{B}}^{-1}_{\infty,\infty})$, and $u_3\in L^\infty(0, T; L^{3, \infty})$ or $u_3\in L^\infty(0, T; \dot{B}^{-1+3/p}_{p, q})$ with $3<p, q< \infty$, then $u$ is smooth in $\mathbb{R}^3\times [0, T]$. Our result improves a previous result establishe…
Optimal conditions for $L^\infty$-regularity and a priori estimates for elliptic systems, II: $n(\geq 3)$ components
本文对一般椭圆型方程组的$n(\geq 3)$分支给出了一个Bootstrap过程。结合$L^p$ - $L^q$ -估计，给出了三种著名弱解：$H_0^1$ -解、$L^1$ -解和$L^1_δ$ -解的最优$L^\infty$ -正则性条件.由于$L^p_δ(Ω)$中的线性理论，它也产生了$L^1_δ$ -解的先验估计的最优条件。基于先验估计，我们改进了几类椭圆型方程组的存在性定理。
We establish heat kernel and gradient estimates for the density of kinetic degenerate Kolmogorov stochastic differentia equations. Our results are established under somehow minimal assumptions that guarantee the SDE is weakly well posed.
In this paper we investigate regularity aspects for solutions of the nonlinear parabolic equation $$ u_t= Δu^m, \quad m > 1 $$ usually called the porous medium equation. More precisely, we provide sharp regularity estimates for bounded nonnegative weak solutions along the free boundary $\partial\{u>0\}$, when the equation is universally close to the heat equation. As a consequence, local Lipschitz estimates are also established for this scenario.
$C^{\infty}$ Regularity for the free boundary of one-phase Fractional Laplacian problem
本文研究含分数阶拉普拉斯算子$(-Δ)^s$（$0<s<1$）的单相自由边界问题，将平坦边界的正则性从已有的$C^{1,α}$提升至$C^∞$，推广了$(-Δ)^{1/2}$情形下的相关结果。
Temporal regularity of the solution to the incompressible Euler equations in the end-point critical Triebel-Lizorkin space $F^{d+1}_{1, \infty}(\mathbb{R}^d)$
本文证明不可压缩欧拉方程的解在端点临界Triebel-Lizorkin空间中存在时间不连续性，从而说明该柯西问题不适定，并进一步讨论了相关空间中解的连续性与弱型连续性。
In this paper the regularity of weak solutions and the blow-up criteria of smooth solutions to the micropolar fluid equations on three dimension space are studied in the Lorentz space $L^{p,\infty}(\mathbb{R}^3)$. We obtain that if $u\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$ for $\frac2q+\frac3p\le 1$ with $3<p\le \infty$; or $\nabla u\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$ for $\frac2q+\frac3p\le 2$ with $\frac32<p\le \infty$; or the pressure $P\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$ for $\frac2q+\frac3p\le 2$ with $\frac32<p\le \infty$; or $\nabla P\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$ f…
We consider the conditional regularity of mild solution $v$ to the incompressible Navier-Stokes equations in three dimensions. Let $e \in \mathbb{S}^2$ and $0 < T^\ast < \infty$. J. Chemin and P. Zhang \cite{CP} proved the regularity of $v$ on $(0,T^\ast]$ if there exists $p \in (4, 6)$ such that $$\int_0^{T^\ast}\|v\cdot e\|^p_{\dot{H}^{\frac{1}{2}+\frac{2}{p}}}dt < \infty.$$ J. Chemin, P. Zhang and Z. F. Zhang \cite{CPZ} extended the range of $p$ to $(4, \infty)$. In this article we settle the case $p \in [2, 4]$. Our proof also works for the case $p \in (4,\infty)$.
Improved regularity estimates for Hardy-Hénon-type equations driven by the $\infty$-Laplacian
本文研究带奇异权重与强吸收的 $\infty$-Laplacian Hardy-Hénon 型方程，证明粘性解的显式改进正则性指数与非退化性，并推广零障碍、死核结果及给出 Liouville 型应用。
Smooth solutions to the axially symmetric Navier-Stokes equations obey the following maximum principle:$\|ru_θ(r,z,t)\|_{L^\infty}\leq\|ru_θ(r,z,0)\|_{L^\infty}.$ We first prove the global regularity of solutions if $\|ru_θ(r,z,0)\|_{L^\infty}$ or $ \|ru_θ(r,z,t)\|_{L^\infty(r\leq r_0)}$ is small compared with certain dimensionless quantity of the initial data. This result improves the one in Zhen Lei and Qi S. Zhang \cite{1}. As a corollary, we also prove the global regularity under the assumption that $|ru_θ(r,z,t)|\leq\ |\ln r|^{-3/2},\ \ \forall\ 0<r\leqδ_0\in(0,1/2).$
We study local regularity properties of a weak solution $u$ to the Cauchy problem of the incompressible Navier-Stokes equations. We present a new regularity criterion for the weak solution $u$ satisfying the condition $L^\infty(0,T;L^{3,w}(\mathbb{R}^3))$ without any smallness assumption on that scale, where $L^{3,w}(\mathbb{R}^3)$ denotes the standard weak Lebesgue space. As an application, we conclude that there are at most a finite number of blowup points at any singular time $t$. The condition that the weak Lebesgue space norm of the veclocity field $u$ is bounded in time is encompassing t…
In this paper we construct a solenoidal vector field $\mathbf{u}$ belonging to $W^{2,q}(Ω)\cap W^{1,s}_0(Ω)$, $s \in (1,\infty)$, $q \in (1,n)$, such that $(1+|\mathbf{Du}|)^{p-2}$, $p\in (1,2)\cup(2,\infty)$, does not belong to the Muckenhoupt class $A_\infty(Ω)$. Thus, one cannot use the Korn inequality in weighted Lebesgue spaces to prove the natural regularity of the $p$-Stokes problem.
A short note on maximal regularity for the heat kernel in Besov spaces and a degenerate 3D Keller-Segel system
本文证明在临界Lorentz空间中具有小初值的退化Keller-Segel方程的整体存在性。
We prove that a solution to the 3D Navier-Stokes or MHD equations does not blow up at $t=T$ provided $\displaystyle \limsup_{q \to \infty} \int_{\mathcal{T}_q}^T \|Δ_q(\nabla \times u)\|_\infty \, dt$ is small enough, where $u$ is the velocity, $Δ_q$ is the Littlewood-Paley projection, and $\mathcal T_q$ is a certain sequence such that $\mathcal T_q \to T$ as $q \to \infty$. This improves many existing regularity criteria.
For the Schrödinger operator $-Δ_\rm{g}+V$ on a complete Riemannian manifold with real valued potential $V$ of compact support, we establish a sharp equivalence between Sobolev regularity of $V$ and the existence of finite-order asymptotic expansions as $t\rightarrow 0$ of the relative trace of the Schrödinger heat kernel. As an application, we generalize a result of Sà Barreto and Zworski, concerning the existence of resonances on compact metric perturbations of Euclidean space, to the case of bounded measurable potentials.
In this article, we study regularity criteria for the 3D micropolar fluid equations in terms of one partial derivative of the velocity. It is proved that if \begin{equation*} \int^{T}_{0}\|\partial_{3}u\|^{\frac{2}{1-r}}_{\dot{B}^{-r}_{\infty,\infty}} dt<\infty \quad \text{with} \quad 0< r<1, \end{equation*} then, the solutions of the micropolar fluid equations actually are smooth on $(0, T)$. This improves and extends many previous results.
Regularity estimates in transport equations via heat flow and quantitative differentiation
本研究借助热流和定量微分，对输运方程进行正则性估计，证明了安布罗西奥 - 特雷维桑及迪佩尔纳 - 利翁斯交换子的定量估计，推动对数索伯列夫正则性传播。
Strong time regularity and decay of $L^\infty$ solutions to $2\times 2$ systems of conservation laws
本文研究$2\times2$守恒律系统的$\mathbf L^\infty$解，利用动力学表述证明有限熵解的强时间正则性，并为消失黏性解建立色散型衰减估计。
This article considers nonlocal heat flows into a singular target space. The problem is the parabolic analogue of a stationary problem that arises as the limit of a singularly perturbed elliptic system. It also provides a gradient flow approach to an optimal eigenvalue partition problem that, in light of the work of Caffarelli and Lin, is equivalent to a constrained harmonic mapping problem. In particular, we show that weak solutions of our heat equation converge as time approaches infinity to stationary solutions of this mapping problem. For these weak solutions, we also prove Lipschitz conti…
This paper establishes the global in time existence of classical solutions to the 2D anisotropic Boussinesq equations with vertical dissipation. When only the vertical dissipation is present, there is no direct control on the horizontal derivatives and the global regularity problem is very challenging. To solve this problem, we bound the derivatives in terms of the $L^\infty$-norm of the vertical velocity $v$ and prove that $\|v\|_{L^{r}}$ with $2\le r<\infty$ at any time does not grow faster than $\sqrt{r \log r}$ as $r$ increases. A delicate interpolation inequality connecting $\|v\|_{L^\inf…
This paper is dedicated to the proof of new maximal regularity results involving Besov spaces for the heat equation in the half-space or in bounded or exterior domains of R^n. We strive for time independent a priori estimates in regularity spaces of type L^1(0,T;X) where X stands for some homogeneous Besov space. In the case of bounded domains, the results that we get are similar to those of the whole space or of the half-space. For exterior domains, we need to use mixed Besov norms in order to get a control on the low frequencies. Those estimates are crucial for proving global-in-time results…
In this paper, we consider the energy conservation and regularity of the weak solution $u$ to the Navier-Stokes equations in the endpoint case. We first construct a divergence-free field $u(t,x)$ which satisfies $\lim_{t\to T}\sqrt{T-t}||u(t)||_{BMO}<\infty$ and $\lim_{t\to T}\sqrt{T-t}||u(t)||_{L^\infty}=\infty$ to demonstrate that the Type II singularity is admissible in the endpoint case $u\in L^{2,\infty}(BMO)$. Secondly, we prove that if a suitable weak solution $u(t,x)$ satisfying $||u||_{L^{2,\infty}([0,T];BMO(Ω))}<\infty$ for arbitrary $Ω\subseteq\mathbb{R}^3$ then the local energy equ…
In this paper, we derive regular criteria via pressure or gradient of the velocity in Lorentz spaces to the 3D Navier-Stokes equations. It is shown that a Leray-Hopf weak solution is regular on $(0,T]$ provided that either the norm $\|Π\|_{L^{p,\infty}(0,T; L ^{q,\infty}(\mathbb{R}^{3}))} $ with $ {2}/{p}+{3}/{q}=2$ $({3}/{2}<q<\infty)$ or $\|\nablaΠ\|_{L^{p,\infty}(0,T; L ^{q,\infty}(\mathbb{R}^{3}))} $ with $ {2}/{p}+{3}/{q}=3$ $(1<q<\infty)$ is small. This gives an affirmative answer to a question proposed by Suzuki in [26, Remark 2.4, p.3850]. Moreover, regular conditions in terms of $\nab…
In this paper, we consider the three-dimensional Navier-Stokes equations, and show that if the $\dot B^{-1}_{\infty,\infty}$-norm of the velocity field is sufficiently small, then the solution is in fact classical.
In this paper, we are concerned with regularity of suitable weak solutions of the 3D Navier-Stokes equations in Lorentz spaces. We obtain $\varepsilon$-regularity criteria in terms of either the velocity, the gradient of the velocity, the vorticity, or deformation tensor in Lorentz spaces. As an application, this allows us to extend the result involving Leray's blow up rate in time, and to show that the number of singular points of weak solutions belonging to $ L^{p,\infty}(-1,0;L^{q,l}(\mathbb{R}^{3})) $ and $ {2}/{p}+{3}/{q}=1$ with $3<q<\infty$ and $q\leq l <\infty$ is finite.
Well-Posedness and Regularity of the Heat Equation with Robin Boundary Conditions in the Two-Dimensional Wedge
本文研究了二维楔形中带有Robin边界条件的热方程的适定性与正则性，通过构建数学框架实现了任意高正则性，无需假设楔形开口角的大小。
We prove that a solution to the three-dimensional Boussinesq equations does not blow-up at time T if $\| u_{\le Q}\|_{B^1_{\infty, \infty}}$ is integrable on $(0, T)$, where $u_{\le Q }$ represents the low modes of Littlewood-Paley projection of the velocity $u$.
Denote by $Δ$ the Laplacian and by $Δ_\infty $ the $\infty$-Laplacian. A fundamental inequality is proved for the algebraic structure of $ΔvΔ_\infty v$: for every $v\in C^\infty$, $$\ | { |D^2vDv|^2} - {Δv Δ_\infty v } -\frac12[|D^2v|^2-(Δv)^2]|Dv|^2\ | \le \frac{n-2}2 [|D^2v|^2{|Dv|^2}- |D^2vDv|^2 ]. $$ Based on this, we prove the following results: 1. For any $p$-harmonic functions $u$, $p\in(1,2)\cup(2,\infty)$, we have $$|Du|^{\frac{p-γ}2}Du\in W^{1,2}_{\rm loc},$$ with $γ<\min\{p+\frac{n-1}{n},3+\frac{p-1}{n-1}\}$. As a by-product, when $p\in(1,2)\cup(2,3+\frac2{n-2})$, we reprove the kno…
In this paper, we study the regularity criterion of weak solutions to the three-dimensional (3D) MHD equations. It is proved that the solution $(u,b)$ becomes regular provided that one velocity and one current density component of the solution satisfy% \begin{equation} u_{3}\in L^{\frac{30α}{7α-45}}\left( 0,T;L^{α,\infty }\left( \mathbb{R}^{3}\right) \right) \text{ \ \ \ with \ \ }\frac{45}{7}% \leq α\leq \infty , \label{eq01} \end{equation}% and \begin{equation} j_{3}\in L^{\frac{2β}{2β-3}}\left( 0,T;L^{β,\infty }\left( \mathbb{R}^{3}\right) \right) \text{ \ \ \ with \ \ }\frac{3}{2}\leq β\le…
$C^\infty$ regularity in semilinear free boundary problems
研究 Alt-Phillips 问题中解与自由边界的高阶正则性，证明自由边界一旦达到 $C^{1,α}$ 即为 $C^\infty$，核心在于建立带边界奇异 Hardy 势线性方程的精细正则性估计。
In this paper we study the Cauchy problem for the semilinear heat and Schrödinger equations, with the nonlinear term $ f ( u ) = λ|u|^αu$. We show that low regularity of $f$ (i.e., $α>0$ but small) limits the regularity of any possible solution for a certain class of smooth initial data. We employ two different methods, which yield two different types of results. On the one hand, we consider the semilinear equation as a perturbation of the ODE $w_t= f(w)$. This yields in particular an optimal regularity result for the semilinear heat equation in Hölder spaces. In addition, this approach yields…
We consider the wellposedness of the fractional Navier-Stokes as a generalization of the wellposedness result in Koch-Tataru's paper. An interesting remark is that our result does not contradict to the well-known ill-posedness result for Navier-Stokes with initial data in $\dot B^{-1, \infty}_\infty$ by Bourgain-Pavlovic. In the end, we also discuss the regularity and analyticity in the space variable.
Regularity theory in generalized function algebras of Colombeau type is largely based on the notion of ${\mathcal G}^\infty$-regularity, which reduces to $C^\infty$-regularity when restricted to Schwartz distributions. Surprisingly, in the nonstandard version of the Colombeau algebras, this basic property of ${\mathcal G}^\infty$-regularity does not hold.
$L^\infty$-regularity thresholds for a generalized Gross-Pitaevskii equation: dispersive blow-up vs. nonlinear stability
该研究针对激子-极化激元玻色-爱因斯坦凝聚体相关的广义Gross-Pitaevskii方程，揭示了稳定弛豫机制与色散爆破的竞争产生$L^\infty$正则性阈值，明确了不同光滑度扰动的爆破与稳定性特性，并在调制空间中完成稳定性分析。
We consider a coupled $1D$ heat-wave system which serves as a simplified fluid-structure interaction problem. The system is coupled in two different ways: the first, when the interface does not have mass and the second, when the interface does have mass. We prove an optimal regularity result in Sobolev spaces for both cases. The main idea behind the proof is to reduce the coupled problem to a single nonlocal equation on the interface by using Neummann to Diriclet operator. Furthermore, we show that point mass coupling regularizes the problem and quantify this regularization in the sense of Sob…
We obtain Lipschitz estimates for bounded minimizers of functionals with nonstandard $(p,q)$-growth satisfying the dimension-independent restriction $q<p+2$ with $p \geq 2$. This relation improves existing restrictions when $p \leq N-1$, moreover our result is sharp in the range $N > \frac{p(2+p)}{2} + 1$. The standard Lipschitz regularity takes the form $W^{1,\infty}_{\text{loc}} - W^{1,p}_{\text{loc}}$, whereas we obtain $W^{1,\infty}_{\text{loc}} - L^{\infty}_{\text{loc}}$ regularity estimate and then make use of existing sharp $L^{\infty}_{\text{loc}}$ bounds to obtain the required conclus…
$C^{\infty}$ regularity of the Alt-Phillips Functional for negative powers
本文研究了Alt-Phillips函数负次幂情况下极值点的正则性，证明了自由边界在正则点处为$C^\infty$，并利用线性化算子建立了比较原理。
Existence, uniqueness and $L^2_t (H_x ^2) \cap L^\infty_t (H^1_x) \cap H^1_t (L^2_x) $ regularity of the gradient flow of the Ambrosio-Tortorelli functional
本文研究二维固定参数下 Ambrosio-Tortorelli 泛函的梯度流，借助新的关键 $L^2_t(H_x^2)$ 估计技术，证明其全局存在性、唯一性及所需混合时空正则性。
We prove that the solutions to the initial-value problem for 2-dimensional Schrödinger maps are unique in $C_tL_x^{\infty} \cap L_t^{\infty} (\dot{H}^1_x\cap \dot{H}^2_x)$. For the proof, we follow McGahagan's argument with improving its technical part, combining Yudovich's argument.
We study the regularity of minima of scalar variational integrals of $p$-growth, $1<p<\infty$, in the Heisenberg group.
$C^\infty$ partial regularity of the singular set in the obstacle problem
本文证明了障碍问题中奇异集可被光滑超曲面覆盖（除低维例外集），并应用于Hele-Shaw流，表明自由边界奇异点仅出现在可数个时刻。
We consider the semilinear elliptic equation $-Δu =λf(u)$ in a smooth bounded domain $Ω$ of $R^{n}$ with Dirichielt boundary condition, where $f$ is a $C^{1}$ positive and nondeccreasing function in $[0,\infty)$ such that $\frac{f(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty$. When $Ω$ is an arbitrary domain and $f$ is not necessarily convex, the boundedness of the extremal solution $u^{*}$ is known only for $n= 2$, established by X. Cabré \cite{C1}. In this paper, we prove this for higher dimensions depending on the nonlinearity $f$. In particular, we prove that if $$\frac{1}{2}<β_{-}:=\li…
Bounded H_\infty-calculus for pseudodifferential Douglis-Nirenberg systems of mild regularity
讨论了拟微分Douglis-Nirenberg系统关于复平面闭子扇区的参数椭圆性，并证明了在Sobolev、Besov和Hoelder空间的适当尺度下存在有界H_ \infty -演算.我们也承认非伪微分扰动。应用涉及系统的系数温和Hoelder正则性和广义热弹性板方程。
We prove an $L^p(I,C^α(Ω))$ regularity result for a reaction-diffusion equation with mixed boundary conditions, symmetric $L^\infty$ coefficients and an $L^\infty$ initial condition. We provide explicit control of the $L^p(I,C^α(Ω))$ norm with respect to the data. To prove our result, we first establish $C^α(Ω)$ control of the stationary equation, extending a result by Haller-Dintelmann et al. (2009).
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