相关论文: On geometric estimates for some problems arising f…
In this paper, we investigate the symmetry properties of positive solutions $u$ to a semilinear elliptic equation under mixed Dirichlet-Neumann boundary conditions in symmetric domains. First, we establish a maximum principle tailored to…
Partial Differential Equations (PDEs) are fundamental for modeling physical systems, yet solving them in a generic and efficient manner using machine learning-based approaches remains challenging due to limited multi-input and multi-scale…
Results on unconditional convergence in the Maximum norm for ADI-type methods, such as the Douglas method, applied to the time integration of semilinear parabolic problems are quite difficult to get, mainly when the number of space…
We prove a sharp H\"older continuity estimates of rupture sets for sequences of solutions of the following nonlinear problem with negative exponent $$ \Delta u= \frac{1}{u^p}, \ p>1, \ \mbox{in} \ \Omega .$$ As a consequence, we prove the…
We propose and analyze a numerical method to solve an elliptic transmission problem in full space. The method consists of a variational formulation involving standard boundary integral operators on the coupling interface and an ultra-weak…
We study the connection between the magnetization profiles of models described by a scalar field with marginal interaction term in a bounded domain and the solutions of the so-called Yamabe problem in the same domain, which amounts to…
This work reformulates the complete electrode model of electrical impedance tomography in order to enable more efficient numerical solution. The model traditionally assumes constant contact conductances on all electrodes, which leads to a…
In this paper, we are concerned with the monotonic and symmetric properties of convex solutions Monge-Amp\`ere systems for instance, considering \begin{equation*} \det(D^2u^i)=f^i(x,{\bf u},\nabla u^i), \ 1\leq i\leq m, \end{equation*} over…
The Minimum Enclosing Ball (MEB) problem is one of the most fundamental problems in clustering, with applications in operations research, statistics and computational geometry. In this works, we give the first differentially private (DP)…
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain satisfying the uniform exterior cone condition. We establish existence and uniqueness of continuous solutions of the Dirichlet Problem associated to certain intrinsic nonlinear mean value…
We give a sufficient condition on nonlinearities of an SDE on a compact connected Riemannian manifold $M$ which implies that laws of all solutions converge weakly to the normalized Riemannian volume measure on $M$. This result is further…
Self-assembled membranes of amphiphilic diblock copolymers enable comparisons of cohesiveness with lipid membranes over the range of hydrophobic thicknesses d=3-15 nm. At zero mechanical tension the breakdown potential V_c for polymersomes…
The p-norm often used in stress constrained topology optimisation supposedly mimics a delta function and it is thus characterised by a small length scale and ideally one would also prefer to have the solid-void transition occur over a small…
We consider the problem of learning an $\varepsilon$-optimal policy in a general class of continuous-space Markov decision processes (MDPs) having smooth Bellman operators. Given access to a generative model, we achieve rate-optimal sample…
We study partially segregated elliptic systems through the use of penalized energy functionals. These systems arise from the minimization of Gross-Pitaevskii-type energies that capture the behavior of multi-component ultracold gas mixtures…
This paper addresses the robust stability of a boundary controlled system coupling two partial differential equations (PDEs), namely beam and string equations, in the presence of boundary and in-domain disturbances under the framework of…
The demagnetization field in micromagnetism is given as the gradient of a potential which solves a partial differential equation (PDE) posed in R^d. In its most general form, this PDE is supplied with continuity condition on the boundary of…
In this work, we are interested in solving large linear systems stemming from the Extra-Membrane-Intra (EMI) model, which is employed for simulating excitable tissues at a cellular scale. After setting the related systems of partial…
Recent years have seen the emergence of nonlinear methods for solving partial differential equations (PDEs), such as physics-informed neural networks (PINNs). While these approaches often perform well in practice, their theoretical analysis…
Monte Carlo approximations for random linear elliptic PDE constrained optimization problems are studied. We use empirical process theory to obtain best possible mean convergence rates $O(n^{-\frac{1}{2}})$ for optimal values and solutions,…