A Nonlinear Elliptic PDE on a Bounded Domain

In summary, a nonlinear elliptic PDE is a mathematical equation involving a function of multiple variables and their derivatives. A bounded domain is a limited region where the PDE is being solved. Solving a nonlinear elliptic PDE on a bounded domain has numerous applications and can be approached using various methods such as finite difference, finite element, and spectral methods. The behavior of the solution to a nonlinear elliptic PDE can be more complex and sensitive to changes compared to a linear PDE.
  • #1
Euge
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Let ##D## be a smooth, bounded domain in ##\mathbb{R}^n## and ##f : D \to (0, \infty)## a continuous function. Prove that there exists no ##C^2##-solution ##u## of the nonlinear elliptic problem ##\Delta u^2 = f## in ##D##, ##u = 0## on ##\partial D##.
 
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  • #2
Start with Green's identity ##\int_D \left(\psi \Delta \psi +||\nabla\psi||^2\right) dV=\int_{\partial D}\psi\nabla\psi\cdot dS.##

Substituting ##\psi=u^2,## we see that the lefthand side is strictly positive, but the righthand side would be zero if ##u=0## on the boundary.
 
  • #3
Just adding a comment @Infrared's solution.

Since the two terms in the integrand on the left-hand side is nonnegative, if its integral over ##D## is zero, then both those terms are zero in ##D##. In particular ##fu^2 = 0## in ##D##, forcing ##u = 0## (since ##f## is positive). We get ##f = \Delta u^2 = 0##, a contradiction. Therefore, the integral on the left-hand side is strictly positive.
 
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  • #4
it is not clear enough what ##u=0## on ##\partial D## means. If ##u\in C^2(D)\cap C(\overline D)## then the assertion follows from the maximum principle directly; smoothness of ##\partial D## is not needed
 

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