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We study the equilibrium shape of liquid drops minimizing the fractional perimeter under the action of a potential energy. We prove, with a quantitative estimate, that the small volume minimizers are convex and uniformly close to a ball.
This paper establishes a strict mathematical relationship between an arbitrary continuous function on a compact set and its global minima, like the well-known first order optimality condition for convex and differentiable functions. By…
Randomness is ubiquitous in modern engineering. The uncertainty is often modeled as random coefficients in the differential equations that describe the underlying physics. In this work, we describe a two-step framework for numerically…
We study a doubly nonlinear parabolic problem arising in the modeling of gas transport in pipelines. Using convexity arguments and relative entropy estimates we show uniform bounds and exponential stability of discrete approximations…
We consider existence and stability of an almost periodic solution of the quasilinear system of differential equations with piecewise constant argument of generalized type. The associated linear homogeneous system satisfies exponential…
Swimming droplets are a class of active particles whose motility changes as a function of time due to shrinkage and self-avoidance of their trail. Here we combine experiments and theory to show that our non-Markovian droplet (NMD) model,…
We propose a generic model for thin films and shallow drops of a polar active liquid that have a free surface and are in contact with a solid substrate. The model couples evolution equations for the film height and the local polarization…
The optimal control of a mechanical system is of crucial importance in many realms. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion…
e study the minimal wave speed and the asymptotics of the traveling wave solutions of a competitive Lotka Volterra system. The existence of the traveling wave solutions is derived by monotone iteration. The asymptotic behaviors of the wave…
We study local asymptotics of solutions to fractional elliptic equations at boundary points, under some outer homogeneous Dirichlet boundary condition. Our analysis is based on a blow-up procedure which involves some Almgren type…
For shape optimization problems, governed by elliptic equations with Dirichlet boundary condition and random coefficients, we utilize a penalization technique to get the approximate problem. We consider that uncertainties exists in the…
Dropout is a widely utilized regularization technique in the training of neural networks, nevertheless, its underlying mechanism and its impact on achieving good generalization abilities remain poorly understood. In this work, we derive the…
For solving unsteady hyperbolic conservation laws on cut cell meshes, the so called small cell problem is a big issue: one would like to use a time step that is chosen with respect to the background mesh and use the same time step on the…
We study almost minimizers for the thin obstacle problem with variable H\"older continuous coefficients and zero thin obstacle and establish their $C^{1,\beta}$ regularity on the either side of the thin space. Under an additional assumption…
We introduce a new quantification of nonuniform ellipticity in variational problems via convex duality, and prove higher differentiability and $2d$-smoothness results for vector valued minimizers of possibly degenerate functionals. Our…
We present a variational framework for studying the existence and regularity of solutions to elliptic free boundary problems that do not necessarily minimize energy. As applications, we obtain mountain pass solutions of critical and…
A block decomposition method is proposed for minimizing a (possibly non-convex) continuously differentiable function subject to one linear equality constraint and simple bounds on the variables. The proposed method iteratively selects a…
We derive necessary optimality conditions for minimizers of regular functionals in the calculus of variations under smooth state constraints. In the literature, this classical problem is widely investigated. The novelty of our result lies…
We study the motion of a droplet evolving by mean curvature with volume constraint and contact angle condition on a half space. We prove the existence of a global-in-time weak solution, called the flat flow. A difficulty arises when we…
The ancient Gamow liquid drop model of nuclear energies has had a renewed life as an interesting problem in the calculus of variations: Find a set $\Omega \subset \mathbb R^3$ with given volume A that minimizes the sum of its surface area…