---
tag: C1F7T523G0
kind: concept
name: Elliptic boundary value problem
msc: [35J25, 35J40]
specializes: [TM8VQJWGHZ]
sources:
  - url: https://en.wikipedia.org/wiki/Elliptic_boundary_value_problem
same_as:
  wikidata: Q5365788
  wikipedia: Elliptic boundary value problem
---

## Summary

An elliptic boundary value problem asks for a solution of an elliptic equation in a domain, subject to conditions on the boundary of the domain.
The model case is the Dirichlet problem for the [[Laplacian|0E0B6RA8K0]]: $\Delta u = f$ inside and $u = g$ on the boundary.
In general, the operator must be elliptic and the boundary conditions must satisfy the [[Lopatinskii–Shapiro condition|48MEJEMEXM]].
Such problems have a priori estimates, regularity up to the boundary and a Fredholm solution theory.

## Explanations

### L1 human

Think of the steady temperature in a metal plate whose edge is held at given temperatures.
Inside, heat flow balances out, which gives an elliptic equation; on the edge, the temperature is prescribed.
Together these determine the temperature everywhere.
Elliptic boundary value problems are the general form of this kind of question, for equilibrium states in physics and geometry.