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The Agmon-Douglis-Nirenberg theorem states(s) state(s) that whenever $f\in W^{m,p}(\Omega)$ where $\Omega$ is a bounded open set of class $C^{m+2}$, then there is a unique solution $u\in W^{m+2,p}(\Omega)$ satisfying the equation $$ -\Delta u+u=f,\;\;\text{ on $\Omega$ with $\|u\|_{W^{m+2,p}}\leq C\|f\|_{W^{m,p}}$ }.$$ Is there a generalization of this result to more general domains $\Omega$'s such as convex polygon or domains with piecewise $C^2$ boundaries?

Edit: I forgot to add that $1<p<\infty$.

The Agmon-Douglis-Nirenberg theorem states that whenever $f\in W^{m,p}(\Omega)$ where $\Omega$ is a bounded open set of class $C^{m+2}$, then there is a unique solution $u\in W^{m+2,p}(\Omega)$ satisfying the equation $$ -\Delta u+u=f,\;\;\text{ on $\Omega$ with $\|u\|_{W^{m+2,p}}\leq C\|f\|_{W^{m,p}}$ }.$$ Is there a generalization of this result to more general domains $\Omega$'s such as convex polygon or domains with piecewise $C^2$ boundaries?

Edit: I forgot to add that $1<p<\infty$.

The Agmon-Douglis-Nirenberg theorem(s) state(s) that whenever $f\in W^{m,p}(\Omega)$ where $\Omega$ is a bounded open set of class $C^{m+2}$, then there is a unique solution $u\in W^{m+2,p}(\Omega)$ satisfying the equation $$ -\Delta u+u=f,\;\;\text{ on $\Omega$ with $\|u\|_{W^{m+2,p}}\leq C\|f\|_{W^{m,p}}$ }.$$ Is there a generalization of this result to more general domains $\Omega$'s such as convex polygon or domains with piecewise $C^2$ boundaries?

Edit: I forgot to add that $1<p<\infty$.

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UserA
UserA
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The Agmon-Douglis-Nirenberg theorem states that whenever $f\in W^{m,p}(\Omega)$ where $\Omega$ is a bounded open set of class $C^{m+2}$, then there is a unique solution $u\in W^{m+2,p}(\Omega)$ satisfying the equation $ -\Delta u+u=f$ on $\Omega$ with $$\|u\|_{W^{m+2,p}}\leq C\|f\|_{W^{m,p}}.$$$$ -\Delta u+u=f,\;\;\text{ on $\Omega$ with $\|u\|_{W^{m+2,p}}\leq C\|f\|_{W^{m,p}}$ }.$$ Is there a generalization of this result to more general domains $\Omega$'s such as convex polygon or domains with piecewise $C^2$ boundaries?

Edit: I forgot to add that $1<p<\infty$.

The Agmon-Douglis-Nirenberg theorem states that whenever $f\in W^{m,p}(\Omega)$ where $\Omega$ is a bounded open of class $C^{m+2}$, then there is a unique solution $u\in W^{m+2,p}(\Omega)$ satisfying the equation $ -\Delta u+u=f$ on $\Omega$ with $$\|u\|_{W^{m+2,p}}\leq C\|f\|_{W^{m,p}}.$$ Is there a generalization of this result to more general domains $\Omega$'s such as convex polygon or domains with piecewise $C^2$ boundaries?

The Agmon-Douglis-Nirenberg theorem states that whenever $f\in W^{m,p}(\Omega)$ where $\Omega$ is a bounded open set of class $C^{m+2}$, then there is a unique solution $u\in W^{m+2,p}(\Omega)$ satisfying the equation $$ -\Delta u+u=f,\;\;\text{ on $\Omega$ with $\|u\|_{W^{m+2,p}}\leq C\|f\|_{W^{m,p}}$ }.$$ Is there a generalization of this result to more general domains $\Omega$'s such as convex polygon or domains with piecewise $C^2$ boundaries?

Edit: I forgot to add that $1<p<\infty$.

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UserA
UserA
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Is there a generalization of the Agmon-Douglis-Nirenberg regularity theorem for elliptic equations to domains with corners?

The Agmon-Douglis-Nirenberg theorem states that whenever $f\in W^{m,p}(\Omega)$ where $\Omega$ is a bounded open of class $C^{m+2}$, then there is a unique solution $u\in W^{m+2,p}(\Omega)$ satisfying the equation $ -\Delta u+u=f$ on $\Omega$ with $$\|u\|_{W^{m+2,p}}\leq C\|f\|_{W^{m,p}}.$$ Is there a generalization of this result to more general domains $\Omega$'s such as convex polygon or domains with piecewise $C^2$ boundaries?