• A convexity problem for a semi-linear PDE

We first prove that the largest subsolution $u$ of the problem exists and is star-shaped with respect to all points in $D$. This in turn implies that the free boundary $\partial \{u >0\}$ is a locally Lipschitz graph. Classical results of free boundary are used to show that the free boundary is $C^{1,\alpha}$. From here we prove uniqueness by a scaling argument , a comparison principle and using the smoothness of the free boundary. We then use quasi-concave envelope $u^*$ of $u$ which is a subsolution of the problem but larger than our solution. This contradicts the fact that our solution is the largest one and we conclude that $u^*=u$. ...

Complete Solution

Posted online: 2019-09-22 09:23:05Z by layan El Hajj71

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• Regularity of Homogenized Boundary Data in Periodic Homogenization of Elliptic Systems

In [1], the authors considered the periodic homogenization of second-order elliptic systems in divergence form with oscillating Dirichlet data or Neumann data of first order. They proved that the homogenized boundary data belongs to $W^{1,p}$ for any $1< p< \infty$. In particular, this implies that the boundary layer tails are Hölder continuous of order $\alpha$ for any $\alpha \in (0, 1)$. ...

Partial Solution

Posted online: 2018-09-29 05:04:52Z by Jinping Zhuge154

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• Existence of a (non-linear) map with desired property that is idempotent and is the identity operator on $W^{1,p}_0(\Omega)$.

Here we will show there exists a map $P_\Omega: W^{1,p}_0(D) \rightarrow W^{1,p}_0(\Omega)$ that satisfies the desired properties of the question and 'behaves' like a projector in the sense that $P^2_\Omega = P_\Omega$ and $P = I$ on $W^{1,p}_0(\Omega)$. Yet $P_\Omega$ is only a linear map if $p=2$.

...

Partial Solution

Posted online: 2018-09-27 20:30:34Z by Shane Cooper79

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• Solution

This problem was solved in 2015 by Dipierro snd Karakhanyan. The paper is published in Advances of Mathematics, 2018, vol. 328, 40-81. The authors used a new dichotomy method, boundary Harnack inequalities, and stratification argument. Their proof applies to more general nonlinear problems. ...

Complete Solution

Posted online: 2018-09-26 09:00:17Z by Aram L Karakhanyan97

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• $L^p$ estimates in homogenization of Dirichlet problem for elliptic systems in divergence form

For each $\varepsilon>0$ let $u_\varepsilon$ be the solution to the following Dirichlet problem $$-\nabla \cdot A_\varepsilon(x) \nabla u_\varepsilon (x) =0 \text{ in } D \ \ \text{ and } \ \ u_\varepsilon(x) = g(x,x/ \varepsilon) \text{ on } \partial D,$$ where $D \subset \mathbb{R}^d$ $(d\geq 2)$ is a bounded domain, $g(x, \cdot)$ is $\mathbb{Z}^d$-periodic for any $x\in \partial D$ and the operator is uniformly elliptic. We study the problem under strict convexity of the domain $D$ and $C^\infty$- smoothness of the data involved in the problem. ...

Partial SolutionYear of origin: 2015

Posted online: 2018-07-07 19:14:43Z by Hayk Aleksanyan83

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• Pointwise estimate in homogenization of Dirichlet problem for elliptic systems in divergence form

We consider homogenization of Dirichlet problem for divergence type elliptic operator when the operator is fixed, and the boundary data is oscillating, which is a particular case of the setting introduced in [2]. ...

Partial SolutionYear of origin: 2013

Posted online: 2018-07-07 19:14:27Z by Hayk Aleksanyan103

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• Homogenization of boundary layers for divergence type operators in strictly convex domains

In a bounded domain $D\subset \mathbb{R}^d$ ($d\geq 2$) consider homogenization of Dirichlet problem of the elliptic system $$\begin{cases} -\nabla \cdot A \left( \frac{x}{\varepsilon} \right) \nabla u(x) = 0, & x \in D, \\ u(x) = g \left(x , \frac{x}{\varepsilon} \right), & x \in \partial D \end{cases} \tag{1}$$ where $\varepsilon > 0$ is a small parameter and $A= A^{\alpha \beta } (x) \in M_N(\mathbb{R})$, $x\in \mathbb{R}^d$ is a family of functions indexed by $1\leq \alpha, \beta \leq d$ with values in the set of matrices $M_N( \mathbb{R})$. ...

Partial SolutionYear of origin: 2012

Posted online: 2018-07-07 19:14:12Z by Hayk Aleksanyan110

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