Timeline for Estimate for elliptic problem on continuous functions
Current License: CC BY-SA 3.0
7 events
when toggle format | what | by | license | comment | |
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Jul 22, 2014 at 9:40 | vote | accept | Matthias Ludewig | ||
Jul 17, 2014 at 13:11 | comment | added | Andrew | There is an $L_p$ theory. It states in particular that for $f\in L_p$, $1<p<\infty$ (and smooth enough $V$), solutions of $Tu=f$ belong locally to $W^{2,p}$. And they do exist. Now from embedding theorems it follows (since a continuous $f$ belongs to $L_p$ for any $p>1$) that solutions belong to $C$ and even to $C^1$. | |
Jul 17, 2014 at 12:12 | comment | added | Matthias Ludewig | Could you explain why there is a solution $u \in C_0$? I don't see how the standard theorems assert this. | |
Jul 17, 2014 at 9:27 | comment | added | Andrew | Surjective from that class? Putting $\alpha=0$ suggests it is $C^2$. For $V\equiv1$ from the example above it follows that there is no a function $u$ from $C^2$ s.t. $Tu=f$ in some neighborhood of the origin. If we consider generalized solutions then of course for every continuous $f$ there is a solution of $Tu=f$ (provided $V$ is regular enough). | |
Jul 17, 2014 at 9:26 | history | edited | Andrew | CC BY-SA 3.0 |
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Jul 17, 2014 at 7:43 | comment | added | Matthias Ludewig | This is a counterexample to a proposed estimate, But this doesn't answer the Question about surjectivity, because the natural domain of $T=\Delta +V$ would be the set of functions such that the distributional derivative $Tu\in C^0$, which would include this function $u$. | |
Jul 16, 2014 at 22:15 | history | answered | Andrew | CC BY-SA 3.0 |