If $(M,g)$ is a Riemannian manifold, we have the well-known base preserving   vector bundle isomorphism $TM\widetilde{=}{T}^{*}M$ given by $v\to g(v,-)$ between the tangent $TM$ and the cotangent $T^{*}M$ bundles of $M$. In the present note, we generalize this isomorphism to the one $T^{\left(r\right)}M\widetilde{=}{T}^{r*}M$ between the $r$-th order vector tangent $T^{\left(r\right)}M={\left(J^r{(M,R)}_0\right)}^{*}$ and the $r$-th order cotangent $T^{r*}M=J^r{(M,R)}_0$ bundles of $M$. Next, we describe all base preserving  vector bundle maps $C_M\left(g\right):T^{\left(r\right)}M\to T^{r*}M$ depending on a Riemannian metric $g$ in terms of natural (in $g$) tensor fields on $M$.

For $n=2m\ge 4$, let $\Omega \in {ℝ}^n$ be a bounded smooth domain and $𝒩\subset {ℝ}^L$ a compact smooth Riemannian manifold without boundary. Suppose that $\left\{u_k\right\}\in W^{m,2}(\Omega ,𝒩)$ is a sequence of weak solutions in the critical dimension to the perturbed $m$-polyharmonic maps $$\frac{\mathrm{d}}{\mathrm{d}t}{|}_{t=0}{E}_m\left(\Pi (u+t\xi )\right)=0$$ with ${\Phi }_k\to 0$ in ${\left(W^{m,2}(\Omega ,𝒩)\right)}^{*}$ and $u_k\rightharpoonup u$ weakly in $W^{m,2}(\Omega ,𝒩)$. Then $u$ is an $m$-polyharmonic map. In particular, the space of $m$-polyharmonic maps is sequentially compact for the weak-$W^{m,2}$ topology.

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

We show that the only flow solving the stochastic differential equation (SDE) on $ℝ$ $$\mathrm{d}{X}_t=1_{\{X_{t}gt;0\}}{W}^{+}\left(\mathrm{d}t\right)+1_{\{X_{t}lt;0\}}\mathrm{d}{W}^{-}\left(\mathrm{d}t\right),$$ where $W^{+}$ and $W^{-}$ are two independent white noises, is a coalescing flow we will denote by ${\varphi }^{\pm }$. The flow ${\varphi }^{\pm }$ is a Wiener solution of the SDE. Moreover, $K^{+}=𝖤\left[{\delta }_{{\varphi }^{\pm }}\right|W^{+}]$ is the unique solution (it is also a Wiener solution) of the SDE $$K_{s,t}^{+}f\left(x\right)=f\left(x\right)+{\int }_s^t{K}_{s,u}\left(1_{ℝ}^{+}{f}^{\text{'}}\right)\left(x\right)W^{+}\left(\mathrm{d}u\right)+\frac{1}{2}{\int }_s^t{K}_{s,u}f``\left(x\right)\mathrm{d}u$$ for $slt;t$, $x\in ℝ$ and $f$ a twice continuously differentiable function. A third flow ${\varphi }^{+}$ can be constructed out of the $n$-point motions of $K^{+}$. This flow is coalescing and its $n$-point motion...

We establish the local-in-time existence of a solution to the non-resistive magneto-micropolar fluids with the initial data $u_0\in H^{s-1+\epsilon }$, $w_0\in H^{s-1}$ and $b_0\in H^s$ for $s>\frac{3}{2}$ and any $0<\epsilon <1$. The initial regularity of the micro-rotational velocity $w$ is weaker than velocity of the fluid $u$.

We study the chemotaxis system with singular sensitivity and logistic-type source: $u_t=\Delta u-\chi \nabla ·(u\nabla v/v)+ru-\mu u^k$, $0=\Delta v-v+u$ under the non-flux boundary conditions in a smooth bounded domain $\Omega \subset {ℝ}^n$, $\chi ,r,\mu >0$, $k>1$ and $n\ge 1$. It is shown with $k\in (1,2)$ that the system possesses a global generalized solution for $n\ge 2$ which is bounded when $\chi >0$ is suitably small related to $r>0$ and the initial datum is properly small, and a global bounded classical solution for $n=1$.

We prove an estimate for the difference of two solutions of the Schrödinger map equation for maps from $T^1$ to $S^2.$ This estimate yields some continuity properties of the flow map for the topology of $L^2(T^1,S^2)$, provided one takes its quotient by the continuous group action of $T^1$ given by translations. We also prove that without taking this quotient, for any $t\&gt;0$ the flow map at time $t$ is discontinuous as a map from ${𝒞}^{\infty }(T^1,S^2)$, equipped with the weak topology of $H^{1/2},$ to the space of distributions ${\left({𝒞}^{\infty }(T^1,{ℝ}^3)\right)}^{*}.$ The argument relies...

We construct some nondecreasing quantities associated to the first eigenvalue of $-{\Delta }_{\phi }+cR$ $(c\ge \frac{1}{2}(n-2)/(n-1))$ along the Yamabe flow, where ${\Delta }_{\phi }$ is the Witten-Laplacian operator with a $C^2$ function $\phi$. We also prove a monotonic result on the first eigenvalue of $-{\Delta }_{\phi }+\frac{1}{4}(n/(n-1))R$ along the Yamabe flow. Moreover, we establish some nondecreasing quantities for the first eigenvalue of $-{\Delta }_{\phi }+c{R}^a$ with $a\in (0,1)$ along the Yamabe flow.

We investigate the recovery of $k$-sparse signals using the ${\ell }_1$-${\ell }_2$ minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume $k$-sparse signals $𝐱$ with the prior support $T$ which is composed of $g$ true indices and $b$ wrong...

Suppose M is a noncompact connected n-manifold and ω is a good Radon measure of M with ω(∂M) = 0. Let ℋ(M,ω) denote the group of ω-preserving homeomorphisms of M equipped with the compact-open topology, and ${\mathscr{H}}_E(M,\omega )$ the subgroup consisting of all h ∈ ℋ(M,ω) which fix the ends of M. S. R. Alpern and V. S. Prasad introduced the topological vector space (M,ω) of end charges of M and the end charge homomorphism $c^{\omega }:{\mathscr{H}}_E(M,\omega )\to (M,\omega )$, which measures for each $h\in {\mathscr{H}}_E(M,\omega )$ the mass flow toward ends induced by h. We show that the...

Let $M$ be an $n$-dimensional ($n\ge 2$) simply connected Hadamard manifold. If the radial Ricci curvature of $M$ is bounded from below by $(n-1)k\left(t\right)$ with respect to some point $p\in M$, where $t=d(·,p)$ is the Riemannian distance on $M$ to $p$, $k\left(t\right)$ is a nonpositive continuous function on $(0,\infty )$, then the first $n$ nonzero Neumann eigenvalues of the Laplacian on the geodesic ball $B(p,l)$, with center $p$ and radius $0<l<\infty$, satisfy $$\frac{1}{{\mu }_1}+\frac{1}{{\mu }_2}+\cdots +\frac{1}{{\mu }_n}\ge \frac{l^{n+2}}{(n+2){\int }_0^l{f}^{n-1}\left(t\right)\mathrm{d}t},$$ where $f\left(t\right)$ is the solution to $$\left\{\begin{array}{c}{f}^{\text{'}\text{'}}\left(t\right)+k\left(t\right)f\left(t\right)=0\text{on}(0,\infty ),\hfill \\ f\left(0\right)=0,f^{\text{'}}\left(0\right)=1.\hfill \end{array}\right.$$

Let $X$ be a Stein manifold of complex dimension $n\ge 2$ and $\Omega ⋐X$ be a relatively compact domain with $C^2$ smooth boundary in $X$. Assume that $\Omega$ is a weakly $q$-pseudoconvex domain in $X$. The purpose of this paper is to establish sufficient conditions for the closed range of $\overline{\partial }$ on $\Omega$. Moreover, we study the $\overline{\partial }$-problem on $\Omega$. Specifically, we use the modified weight function method to study the weighted $\overline{\partial }$-problem with exact support in $\Omega$. Our method relies on the $L^2$-estimates by Hörmander (1965) and by Kohn (1973). ...