What is the regularity of weak solutions of the Liouville equation?

In summary, there have been extensive studies on the regularity of weak solutions for the Liouville equation, which is given by (*) where \Omega \subset \mathbb{R}^2 is a bounded domain with smooth boundary. The equation has been studied in depth in the literature, with references to specific theorems such as Theorem 1.2 in [1] and Theorem 5 in [2]. These studies have been conducted by H. Brezis and L.A. Peletier in [1] and Y. C. Kwong and K. M. Mak in [2].
  • #1
Goklayeh
17
0
Could someone give me some hint (or some reference) about the study of regularity of weak solutions of
[tex]
(*)\quad
\begin{cases}
-\Delta u = e^u & \text{in }\Omega\\
u = 0 & \text{on }\partial \Omega
\end{cases}
[/tex]
where [itex]\Omega \subset \mathbb{R}^2[/itex] is a bounded domain with smooth boundary (here (as usual), a weak solution of (*) is a function [itex]u \in H^1_0(\Omega)[/itex] which satisfies [itex]\int_{\Omega}{\nabla u \cdot \nabla \varphi\mathrm{d}x} = \int_{\Omega}{e^u \varphi \mathrm{d}x}\:\: \forall \varphi \in H^1_0(\Omega)[/itex]).

Thank you in advance for your time!
 
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  • #2
This equation is known as the Liouville equation. Regularity of weak solutions has been studied extensively in the literature; see for example Theorem 1.2 in [1] and Theorem 5 in [2].[1] H. Brezis and L.A. Peletier, Asymptotics for elliptic equations involving exponential nonlinearities, Arch. Rational Mech. Anal. 71 (1979), pp. 143-184.[2] Y. C. Kwong and K. M. Mak, On the regularity of weak solutions of the Liouville equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), pp. 257-273.
 

1. What is a nonlinear PDE?

A nonlinear PDE (partial differential equation) is a mathematical equation that describes the relationship between multiple variables and their corresponding partial derivatives. Unlike linear PDEs, which have a simple and linear relationship between the variables, nonlinear PDEs have more complex and nonlinear relationships.

2. What is the importance of studying the regularity of a nonlinear PDE?

The regularity of a nonlinear PDE refers to how smooth and well-behaved the solutions of the PDE are. Understanding the regularity is important because it allows us to determine the existence and uniqueness of solutions, and also provides information about the behavior of the solutions.

3. How is the regularity of a nonlinear PDE determined?

The regularity of a nonlinear PDE is determined by analyzing the smoothness of the coefficients, boundary conditions, and the initial data of the PDE. It also involves studying the behavior of the solutions near singular points and boundaries.

4. What are some examples of regular and irregular nonlinear PDEs?

An example of a regular nonlinear PDE is the heat equation, which has smooth solutions for all initial and boundary conditions. An example of an irregular nonlinear PDE is the Navier-Stokes equations, which have solutions that are only partially smooth and can exhibit chaotic behavior.

5. How does the regularity of a nonlinear PDE affect its numerical solution?

The regularity of a nonlinear PDE can significantly impact the accuracy and stability of its numerical solution. Highly irregular PDEs can be difficult to solve numerically, and may require specialized methods and algorithms to obtain accurate solutions.

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