PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T20:30:51Z http://mathoverflow.net/feeds/question/69164 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69164/pdes-from-vector-fields-whose-inner-product-with-their-vector-laplacian-equals PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field unknown (yahoo) 2011-06-30T06:45:52Z 2011-06-30T20:33:12Z <p>Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases).</p> <p>Let $\nabla^{2}$ be the vector Laplacian. Let $&lt;\cdot,\cdot>$ be inner product between two vectors.</p> <blockquote> <p><strong>Consider the PDE $&lt;{g},{\nabla^2(g)}> = &lt;{g},{g}>$.</strong></p> <p>$(A)$ <strong>What is such a class of equation formally called in the literature (it seems to be inner product of a field with its vector Laplacian)?</strong></p> <p>$(B)$ <strong>What are the solutions to the above pde?</strong></p> <p>$(C)$ <strong>What are the solutions of $g$ if $g_{i}(x_{1},\cdots,x_{n}) \in [0,1]$ $\forall i$?</strong></p> <p>$(D)$ <strong>What are the solutions for the special case $g_{i}(x_{1},\cdots,x_{n}) = g_{i}(x_{i})$?</strong></p> <p>$(E)$ <strong>What happens if I replace $\mathbb{C}^{n}$ by:</strong></p> <blockquote> <blockquote> <p>$(1)$ <strong>a torus $\mathbb{C}^{n}/L$ where $L$ is a lattice</strong></p> <p>$(2)$ <strong>a sphere centered at $(\frac{1}{2}, \frac{1}{2}, \cdots,\frac{1}{2})$ and radius $\frac{\sqrt{n}}{2}$.</strong></p> <p>$(3)$ <strong>a cube given by the $0-1$ combinations of the standard bases $e_{i}$ (or its closest smooth approximation) enclosing the above sphere.</strong></p> </blockquote> </blockquote> <p>$(F)$ <strong>Does anything interesting happen as limit $n\rightarrow\infty$.</strong></p> </blockquote> <p>I feel this is a standard pde. However, since I am not in the math field, I do not know the keywords or whether there are standard solutions? Where should I look for them? </p>