Timeline for What type of boundary (if any) problem for this family of elliptic PDEs? "half boundary"?
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16 events
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| Dec 26, 2017 at 21:29 | comment | added | zasderf | @NawafBou-Rabee, Yes, $L'$ is degenerate, specifically along $(s,0)$ for nonzero $s$. | |
| Dec 26, 2017 at 21:13 | comment | added | Nawaf Bou-Rabee | @zasderf The operator $L'$ is of degenerate elliptic type. No? (PS: please include my handle in your response, otherwise I won't see it.) | |
| Dec 26, 2017 at 20:44 | comment | added | zasderf | In the original space I had $u=0$ on $ \partial \Omega$ but in this spacetime space, as it includes the variable itself, it is essentially as I written in the above question of having only data only prescribed on the "two sides of the rectangle" rather than the top/bottom where the y-axis is understood as time. For attainability, I assumed some conditions on the Markov transition kernel to attain the expected hitting is finite a.s.. | |
| Dec 26, 2017 at 20:38 | comment | added | zasderf | Long story shorter: For a bounded open set $\Omega$ and some stochastic generator $L$ of an autonomous SDE, it is known that solving $Lu=-1$ on $\Omega$ satisfying boundary condition $u = 0$ on $\partial \Omega$ solves $u$ for expected hitting time. In the case of non-autonomous SDE, the Markov transition function is no longer time-homogeneous which traditionally is used to derive the above PDE. To regain the homogeneity, I considered the spacetime coordinates $(t,X_t)$ which is now time-homogeneous. Perhaps surprisingly, $L'u=-1$ for $L'$ on this spacetime space. | |
| Dec 26, 2017 at 19:18 | history | edited | Nawaf Bou-Rabee |
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| Dec 26, 2017 at 19:08 | comment | added | Nawaf Bou-Rabee | If $\partial \Omega_3 \cup \partial \Omega_4$ are attainable, then don't you have to specify boundary conditions on the process? In particular, what does the process do when it hits $\partial \Omega_3 \cup \partial \Omega_4$? | |
| Dec 26, 2017 at 19:06 | comment | added | zasderf | @NawafBou-Rabee: Actually, that is essentially the problem I am trying to solve in that I started with a SDE and now trying to solve its PDE counterpart. I think from what I've done, it is attainable under some assumption on L. | |
| Dec 26, 2017 at 18:29 | comment | added | Nawaf Bou-Rabee | If you represent $u$ stochastically using a Feynman-Kac formula, then you could describe the Dirichlet boundary conditions on $\partial \Omega_1 \cup \partial \Omega_2$ as absorbing, and the remaining ones as unattainable by the process with generator $L$. | |
| Dec 26, 2017 at 17:59 | comment | added | zasderf | Thank you, Fan Zheng. I suppose the point is that contrasting with well-known literature that one needs $u=0$ (say) on the entire boundary $\partial \Omega$ to ensure uniqueness of solution (in the case of uniformly elliptic operator anyway). | |
| Dec 26, 2017 at 16:50 | comment | added | Fan Zheng | Choose $a$ and $b$ such that $\sin a=\sin b=0$. Choose $c$ and $d$ such that $\sinh c$, $\sinh d\neq0$. | |
| Dec 26, 2017 at 15:45 | comment | added | zasderf | @FanZheng, Yes, I meant $u=0$ rather than $Lu=0$ on the boundary - edited. How did you deduce non-uniqueness from your example -the fact it is not a "complete" Dirichlet boundary? | |
| Dec 26, 2017 at 15:14 | history | edited | zasderf | CC BY-SA 3.0 |
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| Dec 26, 2017 at 2:54 | comment | added | Fan Zheng | The solution is definitely not unique. Consider the harmonic function $u=\sin x\sinh y$. | |
| Dec 26, 2017 at 2:53 | comment | added | Fan Zheng | I guess on the boundary you mean $u=0$ instead of $Lu=0$. | |
| Dec 25, 2017 at 16:59 | review | First posts | |||
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| Dec 25, 2017 at 16:54 | history | asked | zasderf | CC BY-SA 3.0 |