New Heintze-Karcher inequality helps understand droplet shapes.
problem Characterize the shape of droplets inside smooth containers.
method Obtained a new form of the Heintze-Karcher inequality for mean convex hypersurfaces with boundary on curved substrates.
result New mathematical inequality aids in understanding droplet shapes.
Capillarity functionals are parameter invariant functionals defined on classes of two-dimensionals parametric surfaces in R3 as the sum of the area integral with an anisotropic term of suitable form. In the class of parametric surfaces with the topological type of S2 and with fixed volume, extremals of capillarity func…
New proofs and inequalities for capillarity problems quantify asymmetries.
problem Quantifying asymmetries in capillarity functionals.
method ABP-type technique, symmetrization, selection-type argument.
result Sharp quantitative inequalities for asymmetries in capillarity problems.
Capillarity functionals are parameter invariant functionals defined on classes of two-dimensional parametric surfaces in R3 as the sum of the area integral and a non homogeneous term of suitable form. Here we consider the case of a class of non homogenous terms vanishing at infinity for which the corresponding capillar…
Study on capillarity minimizers with nonlocal repulsion and gravity, proving existence and nonexistence.
problem Minimizing an energy functional with capillarity, nonlocal repulsion, and gravity.
method Quantitative isoperimetric inequalities applied to capillarity problem in a half-space.
result Existence and nonexistence of minimizers for various nonlocal kernels and masses.
We study the capillarity equation from the global point of view of behavior of its solutions without explicit regard to boundary conditions. We show its solutions to be constrained in ways, that have till now not been characterized in literature known to us.
Derives equilibrium law for Plateau borders in wet soap films and foams.
problem Equilibrium law for Plateau borders in wet foams and films.
method Rigorous derivation using Gauss' capillarity theory, homotopic spanning condition, and effective compactness theorems.
result Sharp regularity properties of energy minimizers for Plateau borders in wet foams and films.
Soap films collapse only if their bulk has negative pressure, forming convex shapes.
problem Understanding soap film behavior and collapse conditions.
method Variational analysis of capillarity theory.
result Soap films collapse only if their bulk has negative pressure, forming convex shapes.
Smoothness of collapsed regions in soap films is proven, indicating wetted singularities.
problem Smoothness of collapsed regions in soap films.
method Study of generalized minimizers in capillarity model.
result Collapsed regions are smooth outside of dimensionally small singular sets.
Proves existence of minimal surfaces with fixed boundary contact angle.
problem Existence of minimal surfaces with fixed boundary contact angle.
method Min-max construction in the spirit of Almgren-Pitts for the capillarity functional.
result Existence of minimal surfaces in a bounded convex subset of R^3 with fixed boundary contact angle.
New minimal surfaces in spheres with complex topologies from capillarity.
problem Constructing minimal surfaces in spheres with rich topologies.
method General construction of embedded minimal and constant mean curvature surfaces in Sn using capillary hypersurfaces. result Non-trivial sphere bundles over various base spaces, including Stiefel manifolds and complex quadrics.
Study sharp inequalities for perimeter functionals in capillarity and convex cones.
problem Quantitative isoperimetric inequalities for perimeter functionals in capillarity and convex cones.
method Derivation of Fuglede-type estimates and application of selection principle.
result Sharp quantitative isoperimetric inequalities in strong and barycentric forms.
Study proves existence of weak mean curvature flow with contact angle.
problem Existence of weak mean curvature flow with prescribed contact angle.
method Compactness theorem for varifolds and Ilmanen's regularization extended to capillarity.
result Existence of weak mean curvature flow with contact angle for general θ. Study on minimizing singular capillary cones with stability and instability results.
problem Minimizing singular capillary cones with free boundary.
method Stability criterion à la Jerison-Savin, Simons-type inequality for convex, homogeneous, symmetric functions of principal curvatures, boundary condition specific to capillary setting.
result Minimizing cones with non-sign-changing mean curvature are flat in dimensions up to 4, and non-trivial axially symmetric cones are unstable in dimensions up to 6.
The distance of an almost constant mean curvature boundary from a finite family of disjoint tangent balls with equal radii is quantitatively controlled in terms of the oscillation of the scalar mean curvature. This result allows one to quantitatively describe the geometry of volume-constrained stationary sets in capill…
Study approximates Plateau's laws using the Allen-Cahn equation.
problem Approximating Plateau's laws with the Allen-Cahn equation.
method Minimizing the Allen-Cahn energy under volume and spanning constraints.
result Energy minimizing solutions approximate Plateau-type singularities.
Study classifies graphs in Euclidean and non-Euclidean spaces with specific curvature conditions.
problem Classifying graphs with prescribed curvature in various spaces.
method Proves rigidity and classification results for graphs in Riemannian manifolds, focusing on R2 and R3. result Provides general splitting theorems for graphs in these settings.
New varifolds with capillary boundary properties studied.
problem Understanding varifolds with specific boundary conditions.
method Introducing a Radon measure on a Grassmannian bundle as a capillary boundary.
result Structural properties, monotonicity inequality, and integral compactness proved.
Surrogate strategies are used widely for uncertainty quantification of groundwater models in order to improve computational efficiency. However, their application to dynamic multiphase flow problems is hindered by the curse of dimensionality, the saturation discontinuity due to capillarity effects, and the time-depende…
Bayesian inference reconstructs external potentials in DFT for many-particle systems.
problem Reconstructing external potentials in classical density-functional theory (DFT) for many-particle systems.
method Combines Bayesian inference with classical DFT to probabilistically reconstruct external potentials.
result Accurately infers external potentials and density profiles with uncertainty quantification.