We prove a convex integration result for theMonge-Ampère system in dimension $d=2$ and arbitrary codimension $k\geq 1$.We achieve flexibility up to the Hölder regularity $\mathcal{C}^{1,\frac{1}{1+4/k}}$, improving the previous $\mathcal{C}^{1,\frac{1}{1+ 6/k}}$regularity threshold that followed from flexibility up to$\mathcal{C}^{1,\frac{1}{1+d(d+1)/k}}$ in[10], valid forany $d,k\geq 1$. Our result agrees with flexibility up to$\mathcal{C}^{1,\frac{1}{5}}$ for $d=2, k=1$ obtained in[2], andit fills the previously uncharted codimension interval $k\in \big ( 1, 6\big)$,whereas flexibility up to $\mathcal{C}^{1,1}$ has been showed to holdin[10] at any $k\geq 6$.As an application, we state an energy scaling bound in the quantitativeimmersability of Riemannian metrics, for the family ofnonlinear energy functionals modeled on the energies of deformations of thin prestrainedfilms in nonlinear elasticity [9].

We consider the Gelfand problem in a bounded smooth domain with the Dirichlet boundary condition. We are interested in the boundedness of the extremal solution. When the dimension $N\ge10$, it is known that a singular extremal solution is obtained for the exponential nonlinearity if the domain is the unit ball. When $3\le N\le 9$, Cabré, Figalli, Ros-Oton and Serra (2020) proved the following surprising result: the extremal solution is bounded if the nonlinearity is positive, nondecreasing, and convex.In this paper, we succeed in generalizing their result to general $m$-convex nonlinearities. Moreover, we give a unified viewpoint on the results of previous studies by considering $m$-convexity. We obtain a closedness result for stable solutions with $m$-convex nonlinearities. As a consequence, we get a Liouville-type result and by using a blow-up argument, we prove the boundedness of extremal solutions.

We continue to study Streater's model on Brownian particle gas in statistical mechanics on bounded domains in ${\bf R}^n$. If $n\leq 2$ and the entropy production is bounded, there is an a priori estimate of the solution global in time. Under additional properties on entropy and entropy production, this solution converges to a unique steady state, strongly and weakly for the models on classical and quantum particles, respectively.

We consider the Laplacian with a non-homogeneous metricin a tubular neighborhood of a compact hyper-surfacein the Euclidean space of arbitrary dimension,subject to Neumann boundary conditions.It is shown that, in the limit of the width of the neighborhood shrinking to zero, the operator converges in a generalizednorm-resolvent sense to an effective Laplace--Beltrami-type operator on the hyper-surface.In this way, we generalize and give an insight intothe convergence of eigenvalues obtained by Yachimura [19].

The aim of this paper is to study the existence and strong instability of normalized ground states for a Schrödinger equation with van der Waals type potentials, that is, two-body potentials with different width (see [6]).\begin{equation*} \begin{cases} \displaystyle-\Delta u=\lambda u +\mu(I_\alpha\ast|u|^p)|u|^{p-2}u +(I_\beta\ast|u|^q)|u|^{q-2}u, & x\in\mathbb{R}^N, \\\displaystyle\int_{\mathbb{R}^N}|u|^2dx=a^2, & u\in H^1(\mathbb{R}^N), \end{cases}\end{equation*}where $N\geq1$, $\mu > 0$, $$\frac{N +\alpha +2}{N}\leq p < q\leq 2_\beta^*= \begin{cases} \displaystyle +\infty, & N=1,2\\ \displaystyle(N +\beta)/(N-2), & N\geq3 \end{cases} ,$$$0 < \beta < \alpha < N$, $I_\alpha$ and $I_\beta$ are the Riesz potential. We use the subcritical approximation, Pohožaev constraint, Schwarz symmetrization rearrangements method to study mass constrained problem, and we obtain a real-valued, positive, radially symmetric, radially non-increasing and normalized ground state in $H^1(\mathbb{R}^N)$ for the problem. Then, we prove the instability by blow up of the ground state solution for a Schrödinger equation with van der Waals type potentials and nonnegative energy. This extends the results on blowing up solutionsfor the nonlocal Schrödinger equation due to Chen and Guo [9]. The study in this paper is the counterpart of Brezis-Nirenberg type problem for Schrödinger equation with van der Waals type potentials in the context of normalized solutions, and seems to be the first contribution regarding existence of normalized ground states for Schrödinger equation with van der Waals type potentials.