Jove
Visualize
Contact Us

Related Concept Videos

Electrostatic Boundary Conditions01:16

Electrostatic Boundary Conditions

605
Consider an external electric field propagating through a homogeneous medium. When the electric field crosses the surface boundary of the medium, it undergoes a discontinuity. The electric field can be resolved into normal and tangential components. The amount by which the field changes at any boundary is given by the difference between the field components above and below the surface boundary.
The surface integral of an electric field is given by Gauss's law in integral form and is related to...
605
Magnetostatic Boundary Conditions01:28

Magnetostatic Boundary Conditions

1.1K
An electric field suffers a discontinuity at a surface charge. Similarly, a magnetic field is discontinuous at a surface current. The perpendicular component of a magnetic field is continuous across the interface of two magnetic mediums. In contrast, its parallel component, perpendicular to the current, is discontinuous by the amount equal to the product of the vacuum permeability and the surface current. Like the scalar potential in electrostatics, the vector potential is also continuous...
1.1K
Boundary Conditions for Current Density01:25

Boundary Conditions for Current Density

967
Current density becomes discontinuous across an interface of materials with different electrical conductivities. The normal component of the current density is continuous across the boundary.
967
Electrostatic Boundary Conditions in Dielectrics01:27

Electrostatic Boundary Conditions in Dielectrics

1.4K
When an electric field passes from one homogeneous medium to another, crossing the boundary between the two mediums imparts a discontinuity in the electric field. This results in electrostatic boundary conditions that depend on the type of mediums the field propagates through.
Consider a case where both the mediums across a boundary are two different dielectric materials. Recall that the electric field and electric displacement are proportional and related through the material's...
1.4K
Conservation of Mass in Finite Cotrol Volume01:16

Conservation of Mass in Finite Cotrol Volume

1.4K
The principle of conservation of mass is a fundamental law in fluid mechanics and is applied using the continuity equation. We apply the concept to a finite control volume to derive the continuity equation.
A system is defined as a collection of unchanging contents, and the conservation of mass states that a system's mass is constant.
1.4K
Continuity Equation01:20

Continuity Equation

1.0K
The total amount of current flowing per unit cross-sectional area is called the current density. Hence, the current passing through a cross-sectional area can be written as the surface integral of the current density.
1.0K

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Regularity for the Boltzmann Equation Conditional to Pressure and Moment Bounds.

Communications in mathematical physics·2025
Same author

Free Boundary Regularity for Almost Every Solution to the Signorini Problem.

Archive for rational mechanics and analysis·2021
See all related articles
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Experiment Video

Updated: Sep 13, 2025

Control of Cell Adhesion using Hydrogel Patterning Techniques for Applications in Traction Force Microscopy
12:26

Control of Cell Adhesion using Hydrogel Patterning Techniques for Applications in Traction Force Microscopy

Published on: January 29, 2022

5.8K

C ∞ regularity in semilinear free boundary problems.

Daniel Restrepo1, Xavier Ros-Oton2,3,4

  • 1Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218 USA.

Mathematische Annalen
|August 4, 2025
PubMed
Summary

This study investigates the Alt-Phillips problem, revealing that initially smooth free boundaries become infinitely smooth. It also establishes higher regularity for solutions, including critical cases.

Keywords:
35B6535J2535J6135R35

More Related Videos

Investigating the Three-dimensional Flow Separation Induced by a Model Vocal Fold Polyp
09:58

Investigating the Three-dimensional Flow Separation Induced by a Model Vocal Fold Polyp

Published on: February 3, 2014

8.6K
Microtensiometer for Confocal Microscopy Visualization of Dynamic Interfaces
08:05

Microtensiometer for Confocal Microscopy Visualization of Dynamic Interfaces

Published on: September 9, 2022

2.5K

Related Experiment Videos

Last Updated: Sep 13, 2025

Control of Cell Adhesion using Hydrogel Patterning Techniques for Applications in Traction Force Microscopy
12:26

Control of Cell Adhesion using Hydrogel Patterning Techniques for Applications in Traction Force Microscopy

Published on: January 29, 2022

5.8K
Investigating the Three-dimensional Flow Separation Induced by a Model Vocal Fold Polyp
09:58

Investigating the Three-dimensional Flow Separation Induced by a Model Vocal Fold Polyp

Published on: February 3, 2014

8.6K
Microtensiometer for Confocal Microscopy Visualization of Dynamic Interfaces
08:05

Microtensiometer for Confocal Microscopy Visualization of Dynamic Interfaces

Published on: September 9, 2022

2.5K

Area of Science:

  • Mathematical analysis
  • Partial differential equations
  • Geometric measure theory

Background:

  • The Alt-Phillips problem involves a semilinear elliptic equation with a power nonlinearity.
  • Understanding the regularity of solutions and free boundaries is crucial for analyzing such problems.
  • Previous studies have explored various aspects of this problem, but higher regularity remains an active research area.

Purpose of the Study:

  • To establish higher regularity for solutions and free boundaries in the Alt-Phillips problem for $\gamma \in (0, 1)$.
  • To demonstrate that if free boundaries are $C^{1,\alpha}$, they are in fact $C^{\infty}$.
  • To investigate the regularity of specific solution components, namely $u/d^{2/(2-\gamma)}$ and $u^{(2-\gamma)/2}$.

Main Methods:

  • Development of fine regularity estimates for solutions to linear equations with boundary-singular Hardy potentials.
  • Analysis of the equation $-\Delta v = \kappa v/d^2$ in $\Omega$, where $d$ is the distance to the boundary and $\kappa \leq 1/4$.
  • Inclusion of the critical case $\kappa = 1/4$, corresponding to $\gamma = 2/3$.

Main Results:

  • The study proves that free boundaries, once $C^{1,\alpha}$, become $C^{\infty}$.
  • It is shown that the quantities $u/d^{2/(2-\gamma)}$ and $u^{(2-\gamma)/2}$ are $C^{\infty}$.
  • The analysis successfully handles the critical Hardy potential case.

Conclusions:

  • The findings contribute to a deeper understanding of the regularity properties of solutions in the Alt-Phillips problem.
  • The results have implications for the analysis of free boundary problems in mathematical physics and engineering.
  • This work advances the regularity theory for elliptic equations with singular potentials.