User tommi brander - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T05:09:25Z http://mathoverflow.net/feeds/user/1445 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/126950/boundary-regularity-of-weighted-p-laplace-equation Boundary regularity of weighted p-Laplace equation Tommi Brander 2013-04-09T09:47:24Z 2013-04-17T12:58:12Z <p>It seems that I need to find some regularity results for the weighted p-Laplace equation, namely $\nabla \cdot (\gamma |\nabla u|^{p-2} \nabla u) = 0$ with Dirichlet boundary values.</p> <p>Suppose that we have a smooth bounded domain $\Omega$ and smooth Dirichlet boundary values. Take the weight $\gamma$ to be smooth and positive in the closure of $\Omega$. Take p in the interval $]1,\infty[.$</p> <p>What can we say about the regularity of the solution u on the boundary $\partial \Omega$?</p> <p>-</p> <p>Some references I have found:</p> <p>An article by Xiangling Fan titled "Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form" states that with certain assumptions on the domain, weight and boundary conditions we have $u \in C^{1,\alpha}(\overline{\Omega})$. The article focuses on the variable exponent case, though, so better results might exist for static $p$.</p> <p>The work of Kaj Nyström et al concerns the boundary behaviour of the p-laplace equation, but they assume the solution is everywhere positive and zero near the part of boundary they are investigating. These assumptions are too strict for my purposes.</p> <p>The lecture notes of Peter Lindqvist seem to focus more on interior regularity and non-weighted case.</p> <p>-</p> <p>I also asked this on math.SE: <a href="http://math.stackexchange.com/questions/364404/regularity-of-weighted-p-laplace-equation-up-to-boundary" rel="nofollow">http://math.stackexchange.com/questions/364404/regularity-of-weighted-p-laplace-equation-up-to-boundary</a></p> http://mathoverflow.net/questions/88528/what-we-mean-by-positive-solution-and-radial-solution-of-any-partial-differentia/127254#127254 Answer by Tommi Brander for What we mean by positive solution and radial solution of any partial differential equation Tommi Brander 2013-04-11T14:46:41Z 2013-04-11T14:46:41Z <p>Without any further context, I suppose a positive solution is one that is greater than zero everywhere in the domain, i.e. for all $x$ it holds that $u(x) > 0$.</p> <p>A radial solution might mean one that is only a function of the norm of the argument, i.e. for all $x$ it holds that $u(x) = u(\|x\|)$.</p> http://mathoverflow.net/questions/35756/decimating-the-infinite-grid-graph/35768#35768 Answer by Tommi Brander for Decimating the infinite grid graph Tommi Brander 2010-08-16T14:50:03Z 2010-08-16T14:50:03Z <p>For an accessible introduction to percolation theory see the course handout at <a href="https://www.jyu.fi/science/muut_yksikot/summerschool/en/history/JSS19/courses/MA/main#ma2-percolation-theory" rel="nofollow">https://www.jyu.fi/science/muut_yksikot/summerschool/en/history/JSS19/courses/MA/main#ma2-percolation-theory</a> by Jeffrey Steif.</p> http://mathoverflow.net/questions/7859/how-to-teach-addition-of-negative-numbers How to teach addition of negative numbers? Tommi Brander 2009-12-05T10:34:19Z 2010-07-07T06:51:14Z <p>I have a friend with dyscalculia and was teaching her some some mathematics (namely, solving a linear equation, simplifying certain expressions, and what (affine linear) functions are).</p> <p>She understood solving equations of the form $ax + b = 0$ by first adding $-b$ to both sides and then diving by $a$. Dealing with negative $a$ and with expression $b - b$ was something of a problem, but I hope she figured it out, also.</p> <p>Adding slightly more complexity created more problems. For example: $2x + 3 = -7$. We subtract 3 from both sides, getting $2x = -7 - 3$. She has great trouble seeing that $-7-3 = -10$.</p> <p>How to communicate and teach the concepts here? I tried using the thermometer analogy, explaining how $a - a = a + (-a) = 0$ and, somewhat poorly, that $-7 - 3 = - (7 + 3) = -10$. How to justify the last attempt in a useful way? What other models or intuitions are there for understanding the negative numbers and particularly summing them?</p> http://mathoverflow.net/questions/30120/functional-analysis-and-its-relation-to-mechanics/30141#30141 Answer by Tommi Brander for Functional Analysis and its relation to mechanics Tommi Brander 2010-07-01T04:48:52Z 2010-07-01T04:48:52Z <p>Hamilton-Jacobi PDE is a formulation of classical mechanics (as far as I understand; I am no expert in physics) and the unique weak solution is found by a certain calculus of variations problem inspired by optimal control theory.</p> <p>Hamilton-Jacobi is also, I think, somewhat related to the Schrödinger equation.</p> http://mathoverflow.net/questions/3973/what-should-be-offered-in-undergraduate-mathematics-thats-currently-not-or-isn/3979#3979 Answer by Tommi Brander for What should be offered in undergraduate mathematics that's currently not (or isn't usually)? Tommi Brander 2009-11-03T18:04:05Z 2009-11-03T18:04:05Z <p>Basics of numerical methods: What computers can and can't do and how they operate in general.</p> http://mathoverflow.net/questions/124394/p-laplacian-equation Comment by Tommi Brander Tommi Brander 2013-04-10T15:37:39Z 2013-04-10T15:37:39Z For a longer answer see math.stackexchange: <a href="http://math.stackexchange.com/questions/306320/singularity-and-degeneracy-of-solution-to-p-laplace-equation" rel="nofollow" title="singularity and degeneracy of solution to p laplace equation">math.stackexchange.com/questions/306320/&hellip;</a> http://mathoverflow.net/questions/75779/inequalities-and-discontinuous-functions Comment by Tommi Brander Tommi Brander 2011-09-18T20:17:33Z 2011-09-18T20:17:33Z When multiplying by y, if y is negative, then the direction of the inequality changes. http://mathoverflow.net/questions/7859/how-to-teach-addition-of-negative-numbers/7866#7866 Comment by Tommi Brander Tommi Brander 2009-12-05T18:07:04Z 2009-12-05T18:07:04Z She does not know which sort of dyscalclia she has. She does not have multiplications memorised, so I'd guess that basal ganglia is where there are problems, at least. She has problems remembering which hand is the right and the left one and, when given a fraction, is denominator above or below the line, and similar issues. (So do I, for that matter.) Does this help with the classification?