Skip to main content
Notice removed Improve details by CommunityBot
Bounty Ended with no winning answer by CommunityBot
added 19 characters in body
Source Link
Jun
  • 303
  • 2
  • 11

For the solution of $$ \begin{cases} \lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \partial \Omega \end{cases} $$ Varadhan proved that $$\lim_{\epsilon \to 0} - \epsilon \log u^\epsilon = \sqrt{2\lambda} \mathrm{dist} (x,\partial \Omega). $$

Can we prove that somethinga similar asymptotic formula holds for $$ \begin{cases} u^\epsilon + \epsilon (-\Delta)^s u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \mathbb R^N \setminus \Omega \end{cases} $$ holds? Herewhere $(-\Delta)^s$ is the fractional Laplacian operator.?

For the solution of $$ \begin{cases} \lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \partial \Omega \end{cases} $$ Varadhan proved that $$\lim_{\epsilon \to 0} - \epsilon \log u^\epsilon = \sqrt{2\lambda} \mathrm{dist} (x,\partial \Omega). $$

Can we prove that something similar for $$ \begin{cases} u^\epsilon + \epsilon (-\Delta)^s u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \mathbb R^N \setminus \Omega \end{cases} $$ holds? Here $(-\Delta)^s$ is the fractional Laplacian.

For the solution of $$ \begin{cases} \lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \partial \Omega \end{cases} $$ Varadhan proved that $$\lim_{\epsilon \to 0} - \epsilon \log u^\epsilon = \sqrt{2\lambda} \mathrm{dist} (x,\partial \Omega). $$

Can we prove that a similar asymptotic formula holds for $$ \begin{cases} u^\epsilon + \epsilon (-\Delta)^s u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \mathbb R^N \setminus \Omega \end{cases} $$ where $(-\Delta)^s$ is the fractional Laplacian operator?

added 1 character in body
Source Link
Jun
  • 303
  • 2
  • 11

For the solution of $$ \begin{cases} \lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \partial \Omega \end{cases} $$ Varadhan proved that $$\lim_{\epsilon \to 0} - \epsilon \log u^\epsilon = \sqrt{2\lambda} \mathrm{dist} (x,\partial \Omega) $$$$\lim_{\epsilon \to 0} - \epsilon \log u^\epsilon = \sqrt{2\lambda} \mathrm{dist} (x,\partial \Omega). $$

Can we prove that something similar for $$ \begin{cases} u^\epsilon + \epsilon (-\Delta)^s u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \mathbb R^N \setminus \Omega \end{cases} $$ holds? Here $(-\Delta)^s$ is the fractional Laplacian.

For the solution of $$ \begin{cases} \lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \partial \Omega \end{cases} $$ Varadhan proved that $$\lim_{\epsilon \to 0} - \epsilon \log u^\epsilon = \sqrt{2\lambda} \mathrm{dist} (x,\partial \Omega) $$

Can we prove that something similar for $$ \begin{cases} u^\epsilon + \epsilon (-\Delta)^s u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \mathbb R^N \setminus \Omega \end{cases} $$ holds? Here $(-\Delta)^s$ is the fractional Laplacian.

For the solution of $$ \begin{cases} \lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \partial \Omega \end{cases} $$ Varadhan proved that $$\lim_{\epsilon \to 0} - \epsilon \log u^\epsilon = \sqrt{2\lambda} \mathrm{dist} (x,\partial \Omega). $$

Can we prove that something similar for $$ \begin{cases} u^\epsilon + \epsilon (-\Delta)^s u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \mathbb R^N \setminus \Omega \end{cases} $$ holds? Here $(-\Delta)^s$ is the fractional Laplacian.

Notice added Improve details by Jun
Bounty Started worth 50 reputation by Jun
edited title
Link
Jun
  • 303
  • 2
  • 11

Varadhan Asymptotic formula for fractional Laplacian

deleted 20 characters in body
Source Link
Jun
  • 303
  • 2
  • 11
Loading
added 24 characters in body
Source Link
Jun
  • 303
  • 2
  • 11
Loading
added 106 characters in body
Source Link
Jun
  • 303
  • 2
  • 11
Loading
Source Link
Jun
  • 303
  • 2
  • 11
Loading