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The incorporation of ceramics into polymers, forming solid composite electrolytes (SCEs) leads to enhanced electrical performance of all-solid-state lithium metal batteries. This is because the dispersed ceramics particles increase the ionic conductivity, while the polymer matrix leads to better contact performance between the electrolyte and the electrode. In this study, we present a model, based on Hybrid Elements Methods, for the time-dependent Li metal and SCE rough interface mechanics, taking into account for the oxide (ceramics) inclusions (using the Equivalent Inclusion method), and the viscoelasticity of the matrix. We study the effect of LLTO particle size, weight concentration, and spatial distribution on the interface mechanical and electrical response. Moreover, considering the viscoelastic spectrum of a real PEO matrix, under a given stack pressure, we investigate the evolution over time of the mechanical and electrical performance of the interface. The presented theoretical/numerical model might be pivotal in tailoring the development of advanced solid state batteries with superior performance; indeed, we found that conditions in the SCE mixture which optimize both the contact resistivity and the interface stability in time.

In this paper, we focus on the sinh-Gordon equation on graphs. We introduce a uniform a priori estimate to define the topological degree for this equation with nonzero prescribed functions on finite, connected and symmetric graphs. Furthermore, we calculate this topological degree case by case and show several existence results. In particular, we prove that the classical sinh-Gordon equation with nonzero prescribed function is always solvable on such graphs.

Let $(V,E)$ be a finite connected graph. We are concerned about the Chern-Simons Higgs model $$\Delta u=\lambda e^u(e^u-1)+f, \quad\quad\quad\quad\quad\quad{(0.1)}$$ where $\Delta$ is the graph Laplacian, $\lambda$ is a real number and $f$ is a function on $V$. When $\lambda>0$ and $f=4\pi\sum_{i=1}^N\delta_{p_i}$, $N\in\mathbb{N}$, $p_1,\cdots,p_N\in V$, the equation (0.1) was investigated by Huang, Lin, Yau (Commun. Math. Phys. 377 (2020) 613-621) and Hou, Sun (Calc. Var. 61 (2022) 139) via the upper and lower solutions principle. We now consider an arbitrary real number $\lambda$ and a general function $f$, whose integral mean is denoted by $\overline{f}$, and prove that when $\lambda\overline{f}<0$, the equation $(0.1)$ has a solution; when $\lambda\overline{f}>0$, there exist two critical numbers $\Lambda^\ast>0$ and $\Lambda_\ast<0$ such that if $\lambda\in(\Lambda^\ast,+\infty)\cup(-\infty,\Lambda_\ast)$, then $(0.1)$ has at least two solutions, including one local minimum solution; if $\lambda\in(0,\Lambda^\ast)\cup(\Lambda_\ast,0)$, then $(0.1)$ has no solution; while if $\lambda=\Lambda^\ast$ or $\Lambda_\ast$, then $(0.1)$ has at least one solution. Our method is calculating the topological degree and using the relation between the degree and the critical group of a related functional. Similar method is also applied to the Chern-Simons Higgs system, and a partial result for the multiple solutions of the system is obtained., Comment: 20 pages

For a homotopy class $[u]$ of maps between a closed Riemannian manifold $M$ and a general manifold $N$, we want to find a Dirac-harmonic map with map component in the given homotopy class. Most known results require the index to be nontrivial. When the index is trivial, the few known results are all constructive and produce uncoupled solutions. In this paper, we define a new quantity. As a byproduct of proving the homotopy invariance of this new quantity, we find a new simple proof for the fact that all Dirac-harmonic spheres in surfaces are uncoupled. More importantly, by using the homotopy invariance of this new quantity, we prove the existence of Dirac-harmonic maps from manifolds in the trivial index case. In particular, when the domain is a closed Riemann surface, we prove the short-time existence of the $\alpha$-Dirac-harmonic map flow in the trivial index case. Together with the density of the minimal kernel, we get an existence result for Dirac-harmonic maps from closed Riemann surfaces to K\"ahler manifolds, which extends the previous result of the first and third authors. This establishes a general existence theory for Dirac-harmonic maps in the context of trivial index., Comment: arXiv admin note: substantial text overlap with arXiv:1912.11268; text overlap with arXiv:1705.08935 by other authors

Let $(\Sigma,g)$ be a compact Riemann surface with smooth boundary $\partial\Sigma$, $\Delta_g$ be the Laplace-Beltrami operator, and $h$ be a positive smooth function. Using a min-max scheme introduced by Djadli-Malchiodi (2006) and Djadli (2008), we prove that if $\Sigma$ is non-contractible, then for any $\rho\in(8k\pi,8(k+1)\pi)$ with $k\in\mathbb{N}^\ast$, the mean field equation $$\left\{\begin{array}{lll} \Delta_g u=\rho\frac{he^u}{\int_\Sigma he^udv_g}&{\rm in}&\Sigma\\[1.5ex] u=0&{\rm on}&\partial\Sigma \end{array}\right.$$ has a solution. This generalizes earlier existence results of Ding-Jost-Li-Wang (1999) and Chen-Lin (2003) in the Euclidean domain. Also we consider the corresponding Neumann boundary value problem. If $h$ is a positive smooth function, then for any $\rho\in(4k\pi,4(k+1)\pi)$ with $k\in\mathbb{N}^\ast$, the mean field equation $$\left\{\begin{array}{lll} \Delta_g u=\rho\left(\frac{he^u}{\int_\Sigma he^udv_g}-\frac{1}{|\Sigma|}\right)&{\rm in}&\Sigma\\[1.5ex] \partial u/\partial{\mathbf{v}}=0&{\rm on}&\partial\Sigma \end{array}\right.$$ has a solution, where $\mathbf{v}$ denotes the unit normal outward vector on $\partial\Sigma$. Note that in this case we do not require the surface to be non-contractible.

As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise naturally in the singularity analysis of the mean curvature flow. Recently, Guo \cite{Guo} proved that $n$-dimensional compact self-shrinkers in $\mathbb{R}^{n+1}$ with scalar curvature bounded from above or below by some constant are isometric to the round sphere $\mathbb{S}^n(\sqrt{n})$, which implies that $n$-dimensional compact self-shrinkers in $\mathbb{R}^{n+1}$ with constant scalar curvature are isometric to the round sphere $\mathbb{S}^n(\sqrt{n})$(see also \cite{Hui1}). Complete classifications of $n$-dimensional translating solitons in $\mathbb{R}^{n+1}$ with nonnegative constant scalar curvature and of $n$-dimensional self-expanders in $\mathbb{R}^{n+1}$ with nonnegative constant scalar curvature were given by Mart\'{i}n, Savas-Halilaj and Smoczyk\cite{MSS} and Ancari and Cheng\cite{AC}, respectively. In this paper we give complete classifications of $n$-dimensional complete self-shrinkers in $\mathbb{R}^{n+1}$ with nonnegative constant scalar curvature. We will also give alternative proofs of the classification theorems due to Mart\'{i}n, Savas-Halilaj and Smoczyk \cite{MSS} and Ancari and Cheng\cite{AC}., Comment: 15 pages. We noted that Theorem 1.8 in our paper has been proved earlier by Ancari and Cheng in a recent paper [AC]

We study Kazdan-Warner equations on a connected finite graph via the method of the degree theory. Firstly, we prove that all solutions to the Kazdan-Warner equation with nonzero prescribed function are uniformly bounded and the Brouwer degree is well defined. Secondly, we compute the Brouwer degree case by case. As consequences, we give new proofs of some known existence results for the Kazdan-Warner equation on a connected finite graph., Comment: 23 pages

Let $\Sigma$ be a closed Riemann surface, $h$ a positive smooth function on $\Sigma$, $\rho$ and $\alpha$ real numbers. In this paper, we study a generalized mean field equation \begin{align*} -\Delta u=\rho\left(\dfrac{he^u}{\int_\Sigma he^u}-\dfrac{1}{\mathrm{Area}\left(\Sigma\right)}\right)+\alpha\left(u-\fint_{\Sigma}u\right), \end{align*} where $\Delta$ denotes the Laplace-Beltrami operator. We first derive a uniform bound for solutions when $\rho\in (8k\pi, 8(k+1)\pi)$ for some non-negative integer number $k\in \mathbb{N}$ and $\alpha\notin\mathrm{Spec}\left(-\Delta\right)\setminus\set{0}$. Then we obtain existence results for $\alpha<\lambda_1\left(\Sigma\right)$ by using the Leray-Schauder degree theory and the minimax method, where $\lambda_1\left(\Sigma\right)$ is the first positive eigenvalue for $-\Delta$.

In this paper, we study the following Kazdan-Warner equation with sign-changing prescribed function $h$ \begin{align*} -\Delta u=8\pi\left(\frac{he^{u}}{\int_{\Sigma}he^{u}}-1\right) \end{align*} on a closed Riemann surface whose area is equal to one. The solutions are the critical points of the functional $J_{8\pi}$ which is defined by \begin{align*} J_{8\pi}(u)=\frac{1}{16\pi}\int_{\Sigma}|\nabla u|^2+\int_{\Sigma}u-\ln\left|\int_{\Sigma}he^{u}\right|,\quad u\in H^1\left(\Sigma\right). \end{align*} We prove the existence of minimizer of $J_{8\pi}$ by assuming \begin{equation*} \Delta \ln h^++8\pi-2K>0 \end{equation*}at each maximum point of $2\ln h^++A$, where $K$ is the Gaussian curvature, $h^+$ is the positive part of $h$ and $A$ is the regular part of the Green function. This generalizes the existence result of Ding, Jost, Li and Wang [Asian J. Math. 1(1997), 230-248] to the sign-changing prescribed function case. We are also interested in the blow-up behavior of a sequence $u_{\varepsilon}$ of critical points of $J_{8\pi-\varepsilon}$ with $\int_{\Sigma}he^{u_{\varepsilon}}=1, \lim\limits_{\varepsilon\searrow 0}J_{8\pi-\varepsilon}\left(u_{\varepsilon}\right)<\infty$ and obtain the following identity during the blow-up process \begin{equation*} -\varepsilon=\frac{16\pi}{(8\pi-\varepsilon)h(p_\varepsilon)}\left[\Delta \ln h(p_\varepsilon)+8\pi-2K(p_\varepsilon)\right]\lambda_{\varepsilon}e^{-\lambda_{\varepsilon}}+O\left(e^{-\lambda_{\varepsilon}}\right), \end{equation*}where $p_\varepsilon$ and $\lambda_\varepsilon$ are the maximum point and maximum value of $u_\varepsilon$, respectively. Moreover, $p_{\varepsilon}$ converges to the blow-up point which is a critical point of the function $2\ln h^{+}+A$.