Uniqueness of weak solution L[u]=0 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T10:15:45Z http://mathoverflow.net/feeds/question/58583 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58583/uniqueness-of-weak-solution-lu0 Uniqueness of weak solution L[u]=0 Marie 2011-03-15T22:53:46Z 2011-03-18T21:55:48Z <p>Suppose L is a partial differential operator of arbitrary order with constant coefficients.</p> <p>If u is in $L^p(\mathbb{R}^n)$ and Lu=0 in distributions, is it necessarily the case that u=0? Does the answer depend on p? </p> <p>Also, if u is a compactly supported distribution in $\mathbb{R}^n$ with Lu=0 (in the usual sense, i.e. strongly), is it necessarily the case that u=0?</p> <p>(Suggested reference material appreciated) </p> http://mathoverflow.net/questions/58583/uniqueness-of-weak-solution-lu0/58588#58588 Answer by Michael Renardy for Uniqueness of weak solution L[u]=0 Michael Renardy 2011-03-16T00:02:32Z 2011-03-16T02:18:35Z <p>If $Lu=0$, then the Fourier transform of $u$ must have its support on the manifold where the symbol of $L$ is zero. Hence the Fourier transform of $u$ cannot be a function. This rules out $u\in L^p$ for $p\le 2$; it also rules out a compactly supported distribution. The Bessel function $J_0(\sqrt{x^2+y^2})$ satisfies $\Delta u+u=0$, and it is in $L^p$ for every $p>4$. You can generalize this example to $n$ dimensions, and you find that $u\in L^p$ for every $p>2n/(n-1)$.</p> http://mathoverflow.net/questions/58583/uniqueness-of-weak-solution-lu0/58625#58625 Answer by Denis Serre for Uniqueness of weak solution L[u]=0 Denis Serre 2011-03-16T09:28:44Z 2011-03-16T11:58:35Z <p>Strichartz estimates show that the solution $u$ of the Cauchy problem for several equations of physical interest do belong to an $L^p_t(L^q_x)$ if the initial data is appropriate. When $p=q$, this just means that $u\in L^p$.</p> <p>For instance, consider the wave equation $$\partial_t^2u-\Delta_xu=0,\qquad t\in\mathbb R,x\in\mathbb R^d,$$ in which $n=d+1$. Say that $d\ge3$. Let the initial data be $$u(0,x)=a(x),\qquad \partial_tu(0,x)=b(0,x),$$ where $a\in H^1(\mathbb R^d)$ and $b\in L^2(\mathbb R^d)$. Then $u\in L^p(\mathbb R^{1+d})$ with $$p=2\frac{d+1}{d-2}.$$ There are variants of this result, but this is too a rich topic to be developped here.</p> <p><strong>Edit</strong>. This phenomenon is called a <em>dispersion</em> effect. It is related to the fact that the curvature of the characteristic cone of $L$ (here $\xi_0^2=\xi_1^2+\cdots+\xi_d^2$) is non-zero.</p>