Morrey-type regularity of solutions to parabolic problems with discontinuous data

We consider regular oblique derivative problem in cylinder Q _T ?=????× (0, T), ${\Omega\subset {\mathbb R}^n}$ for uniformly parabolic operator ${{{\mathfrak P}}=D_t- \sum_{i,j=1}^n a^{ij}(x)D_{ij}}$ with VMO principal coefficients. Its unique strong solvability is proved in Manuscr. Math. 203?C220 (2000), when ${{{\mathfrak P}}u\in L^p(Q_T)}$ , ${p\in(1,\infty)}$ . Our aim is to show that the solution belongs to the generalized Sobolev?CMorrey space ${W^{2,1}_{p,\omega}(Q_T)}$ , when ${{{\mathfrak P}}u\in L^{p,\omega} (Q_T)}$ , ${p\in (1, \infty)}$ , ${\omega(x,r):\,{\mathbb R}^{n+1}_+\to {\mathbb R}_+}$ . For this goal an a priori estimate is obtained relying on explicit representation formula for the solution. Analogous result holds also for the Cauchy?CDirichlet problem.     

Euclidean hypersurfaces with genuine deformations in codimension two

We classify hypersurfaces of rank two of Euclidean space ${\mathbb{R}^{n+1}}$ that admit genuine isometric deformations in ${\mathbb{R}^{n+2}}$ . That an isometric immersion ${\hat{f}\colon M^n \to \mathbb{R}^{n+2}}$ is a genuine isometric deformation of a hypersurface ${f\colon M^n\to\mathbb{R}^{n+1}}$ means that ${\hat f}$ is nowhere a composition ${\hat f=\hat F\circ f}$ , where ${\hat{F} \colon V\subset \mathbb{R}^{n+1} \to\mathbb{R}^{n+2}}$ is an isometric immersion of an open subset V containing the hypersurface.     

Geometric flows and differential Harnack estimates for heat equations with potentials

Let $M$ be a closed Riemannian manifold with a Riemannian metric $g_{ij}(t)$ evolving by a geometric flow $\partial _{t}g_{ij} = -2{S}_{ij}$ , where $S_{ij}(t)$ is a symmetric two-tensor on $(M, g(t))$ . Suppose that $S_{ij}$ satisfies the tensor inequality $2{\mathcal H}(S, X)+{\mathcal E}(S,X) \ge 0$ for all vector fields $X$ on $M$ , where ${\mathcal H}(S, X)$ and ${\mathcal E}(S,X)$ are introduced in Definition 1 below. Then, we shall prove differential Harnack estimates for positive solutions to time-dependent forward heat equations with potentials. In the case where $S_{ij} = R_{ij}$ , the Ricci tensor of $M$ , our results correspond to the results proved by Cao and Hamilton (Geom Funct Anal 19:983–989, 2009). Moreover, in the case where the Ricci flow coupled with harmonic map heat flow introduced by Müller (Ann Sci Ec Norm Super 45(4):101–142, 2012), our results derive new differential Harnack estimates. We shall also find new entropies which are monotone under the above geometric flow.     

Compact almost Ricci solitons with constant scalar curvature are gradient

The aim of this note is to prove that any compact non-trivial almost Ricci soliton $\big (M^n,\,g,\,X,\,\lambda \big )$ with constant scalar curvature is isometric to a Euclidean sphere $\mathbb {S}^{n}$ . As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreover, the vector field $X$ decomposes as the sum of a Killing vector field $Y$ and the gradient of a suitable function.     

Maximal, potential and singular operators in vanishing generalized Morrey spaces

We introduce vanishing generalized Morrey spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}$ with a general function ${\varphi(x, r)}$ defining the Morrey-type norm. Here ${\Pi \subseteq \Omega}$ is an arbitrary subset in Ω including the extremal cases ${\Pi = \{x_0\}, x_0 \in \Omega}$ and Π = Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n)}$ we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n) \rightarrow V\mathcal{L}^{q,\varphi^\frac{q}{p}}_\Pi (\mathbb{R}^n)}$ -theorem for the potential operator I ^α . The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on ${\varphi(x, r)}$ . No monotonicity type condition is imposed on ${\varphi(x, r)}$ . In case ${\varphi}$ has quasi- monotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function ${\varphi}$ . The proofs are based on pointwise estimates of the modulars defining the vanishing spaces     

Uniqueness properties of degenerate elliptic operators

Let ?? be an open subset of R ^d and ${ K=-\sum^d_{i,j=1}\partial_i\,c_{ij}\,\partial_j+\sum^d_{i=1}c_i\partial_i+c_0}$ a second-order partial differential operator with real-valued coefficients ${c_{ij}=c_{ji}\in W^{1,\infty}_{\rm loc}(\Omega),c_i,c_0\in L_{\infty,{\rm loc}}(\Omega)}$ satisfying the strict ellipticity condition ${C=(c_{ij}) >0 }$ . Further let ${H=-\sum^d_{i,j=1} \partial_i\,c_{ij}\,\partial_j}$ denote the principal part of K. Assuming an accretivity condition ${C\geq \kappa (c\otimes c^{\,T})}$ with ${\kappa >0 }$ , an invariance condition ${(1\!\!1_\Omega, K\varphi)=0}$ and a growth condition which allows ${\|C(x)\|\sim |x|^2\log |x|}$ as |x| ?? ?? we prove that K is L ₁-unique if and only if H is L ₁-unique or Markov unique.     

A Note on Classical and p-adic Fréchet Functional Equations with Restrictions

Given X,Y two ${\mathbb{Q}}$ -vector spaces, and f : X → Y, we study under which conditions on the sets ${B_{k} \subseteq X, k=1,\ldots,s}$ , if ${\Delta_{h_1h_2 \cdots h_s}f(x) = 0}$ for all ${x \in X}$ and ${h_k \in B_k, k = 1,2,\ldots,s}$ , then ${\Delta_{h_1h_2\cdots h_{s}}f(x) = 0}$ for all ${(x,h_{1},\ldots,h_{s}) \in X^{s+1}}$ .     

The Generalization of the Decomposition of Functions by Energy Operators

This work starts with the introduction of a family of differential energy operators. Energy operators $({\varPsi}_{R}^{+}, {\varPsi}_{R}^{-})$ were defined together with a method to decompose the wave equation in a previous work. Here the energy operators are defined following the order of their derivatives $(\varPsi^{-}_{k}, \varPsi^{+}_{k}, k=\{0,\pm 1,\pm 2,\ldots\})$ . The main part of the work demonstrates for any smooth real-valued function f in the Schwartz space $(\mathbf{S}^{-}(\mathbb{R}))$ , the successive derivatives of the n-th power of f ( $n \in \mathbb{Z}$ and n≠0) can be decomposed using only $\varPsi^{+}_{k}$ (Lemma); or if f in a subset of $\mathbf{S}^{-}(\mathbb{R})$ , called $\mathbf{s}^{-}(\mathbb{R})$ , $\varPsi^{+}_{k}$ and $\varPsi^{-}_{k}$ ( $k\in \mathbb{Z}$ ) decompose in a unique way the successive derivatives of the n-th power of f (Theorem). Some properties of the Kernel and the Image of the energy operators are given along with the development. Finally, the paper ends with the application to the energy function.     

Real hypersurfaces of a complex space form

In this paper, we show that an n-dimensional connected non-compact Ricci soliton isometrically immersed in the flat complex space form ${(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle )}$ , with potential vector field of the Ricci soliton is the characteristic vector field of the real hypersurface is an Einstein manifold. We classify connected Hopf hypersurfaces in the flat complex space form ${(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle )}$ and also obtain a characterization for the Hopf hypersurfaces in ${(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle ) }$ .     

Central limit theorems for hyperbolic spaces and Jacobi processes on [0,\infty [

We present a unified approach to a couple of central limit theorems for the radial behavior of radial random walks on hyperbolic spaces as well as for time-homogeneous Markov chains on $[0,\infty [$ whose transition probabilities are defined in terms of Jacobi convolutions. The proofs of all central limit theorems are based on corresponding limit results for the associated Jacobi functions $\varphi _{\lambda }^{(\alpha ,\beta )}$ . In particular, we consider the limit $\alpha \rightarrow \infty $ , the limit $\varphi _{i\rho -n\lambda }^{(\alpha ,\beta )}(t/n)$ for $n\rightarrow \infty $ , and the behavior of the Jacobi function $\varphi _{i\rho -\lambda }^{(\alpha ,\beta )}(t)$ for small $\lambda $ . The proofs of all these limit results are based on the known Laplace integral representation for Jacobi functions. Parts of the results are known, other improve known ones, and other are new.     

Triangles and groups via cevians

For a given triangle T and a real number ρ we define Ceva’s triangle ${\mathcal{C}_{\rho}(T)}$ to be the triangle formed by three cevians each joining a vertex of T to the point which divides the opposite side in the ratio ρ: (1 – ρ). We identify the smallest interval ${\mathbb{M}_T \subset \mathbb{R}}$ such that the family ${\mathcal{C}_{\rho}(T), \rho \in \mathbb{M}_T}$ , contains all Ceva’s triangles up to similarity. We prove that the composition of operators ${\mathcal{C}_\rho, \rho \in \mathbb{R}}$ , acting on triangles is governed by a certain group structure on ${\mathbb{R}}$ . We use this structure to prove that two triangles have the same Brocard angle if and only if a congruent copy of one of them can be recovered by sufficiently many iterations of two operators ${\mathcal{C}_\rho}$ and ${\mathcal{C}_\xi}$ acting on the other triangle.     

Tiling Convex Polygons with Congruent Equilateral Triangles

We study the sets $\mathcal{T}_{v}=\{m \in\{1,2,\ldots\}: \mbox{there is a convex polygon in }\mathbb{R}^{2}\mbox{ that has }v\mbox{ vertices and can be tiled with $m$ congruent equilateral triangles}\}$ , v=3,4,5,6. $\mathcal{T}_{3}$ , $\mathcal{T}_{4}$ , and $\mathcal{T}_{6}$ can be quoted completely. The complement $\{1,2,\ldots\} \setminus\mathcal{T}_{5}$ of $\mathcal{T}_{5}$ turns out to be a subset of Euler’s numeri idonei. As a consequence, $\{1,2,\ldots\} \setminus\mathcal{T}_{5}$ can be characterized with up to two exceptions, and a complete characterization is given under the assumption of the Generalized Riemann Hypothesis.     

Lax embeddings of the Hermitian unital

In this paper, we prove that every lax generalized Veronesean embedding of the Hermitian unital ${\mathcal{U}}$ of ${\mathsf{PG}(2,\mathbb{L}), \mathbb{L}}$ a quadratic extension of the field ${\mathbb{K}}$ and ${|\mathbb{K}| \geq 3}$ , in a ${\mathsf{PG}(d,\mathbb{F})}$ , with ${\mathbb{F}}$ any field and d ≥ 7, such that disjoint blocks span disjoint subspaces, is the standard Veronesean embedding in a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ (and d = 7) or it consists of the projection from a point ${p \in \mathcal{U}}$ of ${\mathcal{U}{\setminus} \{p\}}$ from a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ into a hyperplane ${\mathsf{PG}(6,\mathbb{K}^{\prime})}$ . In order to do so, when ${|\mathbb{K}| >3 }$ we strongly use the linear representation of the affine part of ${\mathcal{U}}$ (the line at infinity being secant) as the affine part of the generalized quadrangle ${\mathsf{Q}(4,\mathbb{K})}$ (the solid at infinity being non-singular); when ${|\mathbb{K}| =3}$ , we use the connection of ${\mathcal{U}}$ with the generalized hexagon of order 2.     

Harmonic mappings and conformal minimal immersions of Riemann surfaces into {\mathbb {R}^{\rm N}}

We prove that for any open Riemann surface ${\mathcal{N}}$ , natural number N ≥ 3, non-constant harmonic map ${h:\mathcal{N} \to \mathbb{R}}$ ^N?2 and holomorphic 2-form ${\mathfrak{H}}$ on ${\mathcal{N}}$ , there exists a weakly complete harmonic map ${X=(X_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}$ with Hopf differential ${\mathfrak{H}}$ and ${(X_j)_{j=3,\ldots,{\sc N}}=h.}$ In particular, there exists a complete conformal minimal immersion ${Y=(Y_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}$ such that ${(Y_j)_{j=3,\ldots,{\sc N}}=h}$ . As some consequences of these results (1) there exist complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss map is non-degenerate and fails to intersect N hyperplanes of ${\mathbb{CP}^{{\sc N}-1}}$ in general position. (2) There exist complete non-proper embedded minimal surfaces in ${\mathbb{R}^{\sc N},}$ ${\forall\,{\sc N} >3 .}$      

Growth estimates for orbits of self-adjoint groups

Let $G$ denote a closed, connected, self-adjoint, noncompact subgroup of $GL(n,\mathbb R )$ , and let $d_{R}$ and $d_{L}$ denote respectively the right and left invariant Riemannian metrics defined by the canonical inner product on $M(n,\mathbb R ) = T_{I} GL(n,\mathbb R )$ . Let $v$ be a nonzero vector of $\mathbb R ^{n}$ such that the orbit $G(v)$ is unbounded in $\mathbb R ^{n}$ . Then the function $g \rightarrow d_{R}(g, G_{v})$ is unbounded, where $G_{v} = \{g \in G : g(v) = v \}$ , and we obtain algebraically defined upper and lower bounds $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ for the asymptotic behavior of the function $\frac{log|g(v)|}{d_{R}(g, G_{v})}$ as $d_{R}(g, G_{v}) \rightarrow \infty $ . The upper bound $\lambda ^{+}(v)$ is at most 1. The orbit $G(v)$ is closed in $\mathbb R ^{n} \Leftrightarrow \lambda ^{-}(w)$ is positive for some w $\in G(v)$ . If $G_{v}$ is compact, then $g \rightarrow |d_{R}(g,I) - d_{L}(g,I)|$ is uniformly bounded in $G$ , and the exponents $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ are sharp upper and lower asymptotic bounds for the functions $\frac{log|g(v)|}{d_{R}(g,I)}$ and $\frac{log|g(v)|}{d_{L}(g,I)}$ as $d_{R}(g,I) \rightarrow \infty $ or as $d_{L}(g,I) \rightarrow \infty $ . However, we show by example that if $G_{v}$ is noncompact, then there need not exist asymptotic upper and lower bounds for the function $\frac{log|g(v)|}{d_{L}(g, G_{v})}$ as $d_{L}(g, G_{v}) \rightarrow \infty $ . The results apply to representations of noncompact semisimple Lie groups $G$ on finite dimensional real vector spaces. We compute $\lambda ^{+}$ and $\lambda ^{-}$ for the irreducible, real representations of $SL(2,\mathbb R )$ , and we show that if the dimension of the $SL(2,\mathbb R )$ -module $V$ is odd, then $\lambda ^{+} = \lambda ^{-}$ on a nonempty open subset of $V$ . We show that the function $\lambda ^{-}$ is $K$ -invariant, where $K = O(n,\mathbb R ) \cap G$ . We do not know if $\lambda ^{-}$ is $G$ -invariant.     

On the singular set of minimizers of p(x)-energies

We treat the partial regularity of locally bounded local minimizers $u$ for the $p(x)$ -energy functional $$\begin{aligned} \mathcal{E }(v;\Omega ) = \int \left( g^{\alpha \beta }(x)h_{ij}(v) D_\alpha v^i (x) D_\beta v^j (x) \right) ^{p(x)/2} dx, \end{aligned}$$ defined for maps $v : \Omega (\subset \mathbb R ^m) \rightarrow \mathbb R ^n$ . Assuming the Lipschitz continuity of the exponent $p(x) \ge 2$ , we prove that $u \in C^{1,\alpha }(\Omega _0)$ for some $\alpha \in (0,1)$ and an open set $\Omega _0 \subset \Omega $ with $\dim _\mathcal{H }(\Omega \setminus \Omega _0) \le m-[\gamma _1]-1$ , where $\dim _\mathcal{H }$ stands for the Hausdorff dimension, $[\gamma _1]$ the integral part of $\gamma _1$ , and $\gamma _1 = \inf p(x)$ .     

Finite-time blowup for a complex Ginzburg–Landau equation with linear driving

In this paper, we consider the complex Ginzburg–Landau equation ${u_t = e^{i\theta} [\Delta u + |u|^\alpha u] + \gamma u}$ on ${\mathbb{R}^N}$ , where ${\alpha > 0,\,\gamma \in \mathbb{R}}$ and ${-\pi /2 < \theta < \pi /2}$ . By convexity arguments, we prove that, under certain conditions on ${\alpha,\theta,\gamma}$ , a class of solutions with negative initial energy blows up in finite time.     

A Weighted Sobolev Space Theory of Parabolic Stochastic PDEs on Non-smooth Domains

In this article, we study parabolic stochastic partial differential equations (see Eq. (1.1)) defined on arbitrary bounded domain $\mathcal{O }\subset \mathbb{R }^d$ admitting the Hardy inequality 0.1 $$\begin{aligned} \int _{\mathcal{O }}|\rho ^{-1}g|^2\,\text{ d}x\le C\int _{\mathcal{O }}|g_x|^2 \text{ d}x, \quad \forall g\in C^{\infty }_0(\mathcal{O }), \end{aligned}$$ where $\rho (x)=\text{ dist}(x,\partial \mathcal{O }).$ Existence and uniqueness results are given in weighted Sobolev spaces $\mathfrak{H }_{p,\theta }^{\gamma }(\mathcal{O },T),$ where $p\in [2,\infty ), \gamma \in \mathbb{R }$ is the number of derivatives of solutions and $\theta $ controls the boundary behavior of solutions (see Definition 2.5). Furthermore, several Hölder estimates of the solutions are also obtained. It is allowed that the coefficients of the equations blow up near the boundary.