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Fixed several typos (there were $v$s in the definition of the norms of $u$) and some layout
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edit approved Sep 27, 2022 at 13:20
Hannes
edit approved Sep 27, 2022 at 13:20
Hannes
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Elliptic partial differential equations with robinRobin boundary condition and domain of fractional power of Robin Laplacian operator

This question has been posted on Mathematics Stack Exchange but got no response, and so I put it here.

When I read the paper "On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation" by Igor Chueshov, Matthias Eller and Irena Lasiecka, I encounter a difficulty:

Suppose that $\Omega$ is a simple connected domain with smooth boundary, the authors introduced a Robin Laplacian operator $$\Delta_{R}:L^{2}(\Omega)\to L^{2}(\Omega).$$$$\Delta_{R}\colon L^{2}(\Omega)\to L^{2}(\Omega).$$This is an unbounded operator with the domain $$D(\Delta_{R})=\{u\in H^{2}(\Omega):\partial_{\nu}u+u=0\ on\ \partial\Omega\}.$$Moreover$$D(\Delta_{R})=\bigl\{u\in H^{2}(\Omega)\colon \partial_{\nu}u+u=0\ on\ \partial\Omega\bigr\}.$$ Moreover, the Robin Laplacian can be extended to a continuous operator $\Delta_{R}:H^{1}(\Omega)\to H^{1}(\Omega)'$$\Delta_{R}\colon H^{1}(\Omega)\to H^{1}(\Omega)'$ by $$(-\Delta_{R}u,v)_{L^{2}(\Omega)}=(\nabla u,\nabla v)_{L^{2}(\Omega)}+<u,v>_{L^{2}(\partial\Omega)}.$$$$(-\Delta_{R}u,v)_{L^{2}(\Omega)}=(\nabla u,\nabla v)_{L^{2}(\Omega)}+\langle u,v\rangle_{L^{2}(\partial\Omega)}.$$ Then the authors say "this extension is the duality map $H^{1}(\Omega)$ into $(H^{1}(\Omega))'$" when we equip $H^{1}(\Omega)$ the norm $$\|u\|=\sqrt{(\nabla u,\nabla v)_{L^{2}(\Omega)}+<u,v>_{L^{2}(\partial\Omega)}}.$$$$\|u\|^* =\sqrt{(\nabla u,\nabla u)_{L^{2}(\Omega)}+\langle u,u\rangle_{L^{2}(\partial\Omega)}}.$$ So, firstly, I want to know if this norm is equivalent to the usual Sobolev norm $$\|u\|=\sqrt{(\nabla u,\nabla v)_{L^{2}(\Omega)}+(u,v)_{L^{2}(\Omega)}}$$$$\|u\|=\sqrt{(\nabla u,\nabla u)_{L^{2}(\Omega)}+(u,u)_{L^{2}(\Omega)}}$$ when $u$ satisfies the robinRobin boundary condition? Next, the authors said $$D((-\Delta_{R}))^{\frac{1}{2}}\sim H^{1}(\Omega),$$$$D((-\Delta_{R})^{\frac{1}{2}})\sim H^{1}(\Omega),$$ why? The authors offered a reference but it is French, but I don't understand French. theThe reference is "Grisvard,P P.: Characterisation de Quelques EsoacesEspaces d'interpolation. Archives Rational Mechanics and Analysis 1967,26(1967) #26, pp.40-63"

Any comments and hints are welcome, thank you very much!!!

Elliptic partial differential equations with robin boundary condition and domain of fractional power of Robin Laplacian operator

This question has been posted on Mathematics Stack Exchange but got no response, and so I put it here.

When I read the paper "On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation" by Igor Chueshov, Matthias Eller and Irena Lasiecka, I encounter a difficulty:

Suppose that $\Omega$ is a simple connected domain with smooth boundary, the authors introduced a Robin Laplacian operator $$\Delta_{R}:L^{2}(\Omega)\to L^{2}(\Omega).$$This is an unbounded operator with the domain $$D(\Delta_{R})=\{u\in H^{2}(\Omega):\partial_{\nu}u+u=0\ on\ \partial\Omega\}.$$Moreover, the Robin Laplacian can be extended to a continuous operator $\Delta_{R}:H^{1}(\Omega)\to H^{1}(\Omega)'$ by $$(-\Delta_{R}u,v)_{L^{2}(\Omega)}=(\nabla u,\nabla v)_{L^{2}(\Omega)}+<u,v>_{L^{2}(\partial\Omega)}.$$ Then the authors say "this extension is the duality map $H^{1}(\Omega)$ into $(H^{1}(\Omega))'$" when we equip $H^{1}(\Omega)$ the norm $$\|u\|=\sqrt{(\nabla u,\nabla v)_{L^{2}(\Omega)}+<u,v>_{L^{2}(\partial\Omega)}}.$$ So, firstly, I want to know if this norm is equivalent to the usual Sobolev norm $$\|u\|=\sqrt{(\nabla u,\nabla v)_{L^{2}(\Omega)}+(u,v)_{L^{2}(\Omega)}}$$ when $u$ satisfies the robin boundary condition? Next, the authors said $$D((-\Delta_{R}))^{\frac{1}{2}}\sim H^{1}(\Omega),$$ why? The authors offered a reference but it is French, but I don't understand French. the reference is "Grisvard,P. Characterisation de Quelques Esoaces d'interpolation. Archives Rational Mechanics and Analysis 1967,26,40-63"

Any comments and hints are welcome, thank you very much!!!

Elliptic partial differential equations with Robin boundary condition and domain of fractional power of Robin Laplacian operator

This question has been posted on Mathematics Stack Exchange but got no response, and so I put it here.

When I read the paper "On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation" by Igor Chueshov, Matthias Eller and Irena Lasiecka, I encounter a difficulty:

Suppose that $\Omega$ is a simple connected domain with smooth boundary, the authors introduced a Robin Laplacian operator $$\Delta_{R}\colon L^{2}(\Omega)\to L^{2}(\Omega).$$This is an unbounded operator with the domain $$D(\Delta_{R})=\bigl\{u\in H^{2}(\Omega)\colon \partial_{\nu}u+u=0\ on\ \partial\Omega\bigr\}.$$ Moreover, the Robin Laplacian can be extended to a continuous operator $\Delta_{R}\colon H^{1}(\Omega)\to H^{1}(\Omega)'$ by $$(-\Delta_{R}u,v)_{L^{2}(\Omega)}=(\nabla u,\nabla v)_{L^{2}(\Omega)}+\langle u,v\rangle_{L^{2}(\partial\Omega)}.$$ Then the authors say "this extension is the duality map $H^{1}(\Omega)$ into $(H^{1}(\Omega))'$" when we equip $H^{1}(\Omega)$ the norm $$\|u\|^* =\sqrt{(\nabla u,\nabla u)_{L^{2}(\Omega)}+\langle u,u\rangle_{L^{2}(\partial\Omega)}}.$$ So, firstly, I want to know if this norm is equivalent to the usual Sobolev norm $$\|u\|=\sqrt{(\nabla u,\nabla u)_{L^{2}(\Omega)}+(u,u)_{L^{2}(\Omega)}}$$ when $u$ satisfies the Robin boundary condition? Next, the authors said $$D((-\Delta_{R})^{\frac{1}{2}})\sim H^{1}(\Omega),$$ why? The authors offered a reference but it is French, but I don't understand French. The reference is "Grisvard, P.: Characterisation de Quelques Espaces d'interpolation. Archives Rational Mechanics and Analysis (1967) #26, pp.40-63"

Any comments and hints are welcome, thank you very much!

added 67 characters in body
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Glorfindel
Glorfindel
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This question have been posted in stack exchangehas been posted on Mathematics Stack Exchange but nobody caregot no response, and so I put it here. When

When I read the paper "On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation" by Igor Chueshov,Matthias Matthias Eller and Irena Lasiecka,I I encounter a difficulty: Suppose

Suppose that $\Omega$ is a simple connected domain with smooth boundary, the authors introduced a Robin Laplacian operator $$\Delta_{R}:L^{2}(\Omega)\to L^{2}(\Omega).$$This is an unbounded operator with the domain $$D(\Delta_{R})=\{u\in H^{2}(\Omega):\partial_{\nu}u+u=0\ on\ \partial\Omega\}.$$Moreover, the Robin Laplacian can be extended to a continuous operator $\Delta_{R}:H^{1}(\Omega)\to H^{1}(\Omega)'$ by $$(-\Delta_{R}u,v)_{L^{2}(\Omega)}=(\nabla u,\nabla v)_{L^{2}(\Omega)}+<u,v>_{L^{2}(\partial\Omega)}.$$ Then the authors say "this extension is the duality map $H^{1}(\Omega)$ into $(H^{1}(\Omega))'$" when we equip $H^{1}(\Omega)$ the norm $$\|u\|=\sqrt{(\nabla u,\nabla v)_{L^{2}(\Omega)}+<u,v>_{L^{2}(\partial\Omega)}}.$$ So, firstly, I want to know if this norm is equivalent to the usual Sobolev norm $$\|u\|=\sqrt{(\nabla u,\nabla v)_{L^{2}(\Omega)}+(u,v)_{L^{2}(\Omega)}}$$ when $u$ satisfies the robin boundary conditoncondition? Next, the authors said $$D((-\Delta_{R}))^{\frac{1}{2}}\sim H^{1}(\Omega),$$ why? theThe authors offered ana reference but it is frenchFrench, but I don't understand frenchFrench. the reference is "Grisvard,P. Characterisation de Quelques Esoaces d'interpolation. Archives Rational Mechanics and Analysis 1967,26,40-63"

Any comments and hints are welcome, thank you very much!!!

This question have been posted in stack exchange but nobody care, and so I put it here. When I read the paper "On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation" by Igor Chueshov,Matthias Eller and Irena Lasiecka,I encounter a difficulty: Suppose that $\Omega$ is a simple connected domain with smooth boundary, the authors introduced a Robin Laplacian operator $$\Delta_{R}:L^{2}(\Omega)\to L^{2}(\Omega).$$This is an unbounded operator with the domain $$D(\Delta_{R})=\{u\in H^{2}(\Omega):\partial_{\nu}u+u=0\ on\ \partial\Omega\}.$$Moreover, the Robin Laplacian can be extended to a continuous operator $\Delta_{R}:H^{1}(\Omega)\to H^{1}(\Omega)'$ by $$(-\Delta_{R}u,v)_{L^{2}(\Omega)}=(\nabla u,\nabla v)_{L^{2}(\Omega)}+<u,v>_{L^{2}(\partial\Omega)}.$$ Then the authors say "this extension is the duality map $H^{1}(\Omega)$ into $(H^{1}(\Omega))'$" when we equip $H^{1}(\Omega)$ the norm $$\|u\|=\sqrt{(\nabla u,\nabla v)_{L^{2}(\Omega)}+<u,v>_{L^{2}(\partial\Omega)}}.$$ So, firstly, I want to know if this norm is equivalent to the usual Sobolev norm $$\|u\|=\sqrt{(\nabla u,\nabla v)_{L^{2}(\Omega)}+(u,v)_{L^{2}(\Omega)}}$$ when $u$ satisfies the robin boundary conditon? Next, the authors said $$D((-\Delta_{R}))^{\frac{1}{2}}\sim H^{1}(\Omega),$$ why? the authors offered an reference but it is french, but I don't understand french. the reference is "Grisvard,P. Characterisation de Quelques Esoaces d'interpolation. Archives Rational Mechanics and Analysis 1967,26,40-63"

Any comments and hints are welcome, thank you very much!!!

This question has been posted on Mathematics Stack Exchange but got no response, and so I put it here.

When I read the paper "On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation" by Igor Chueshov, Matthias Eller and Irena Lasiecka, I encounter a difficulty:

Suppose that $\Omega$ is a simple connected domain with smooth boundary, the authors introduced a Robin Laplacian operator $$\Delta_{R}:L^{2}(\Omega)\to L^{2}(\Omega).$$This is an unbounded operator with the domain $$D(\Delta_{R})=\{u\in H^{2}(\Omega):\partial_{\nu}u+u=0\ on\ \partial\Omega\}.$$Moreover, the Robin Laplacian can be extended to a continuous operator $\Delta_{R}:H^{1}(\Omega)\to H^{1}(\Omega)'$ by $$(-\Delta_{R}u,v)_{L^{2}(\Omega)}=(\nabla u,\nabla v)_{L^{2}(\Omega)}+<u,v>_{L^{2}(\partial\Omega)}.$$ Then the authors say "this extension is the duality map $H^{1}(\Omega)$ into $(H^{1}(\Omega))'$" when we equip $H^{1}(\Omega)$ the norm $$\|u\|=\sqrt{(\nabla u,\nabla v)_{L^{2}(\Omega)}+<u,v>_{L^{2}(\partial\Omega)}}.$$ So, firstly, I want to know if this norm is equivalent to the usual Sobolev norm $$\|u\|=\sqrt{(\nabla u,\nabla v)_{L^{2}(\Omega)}+(u,v)_{L^{2}(\Omega)}}$$ when $u$ satisfies the robin boundary condition? Next, the authors said $$D((-\Delta_{R}))^{\frac{1}{2}}\sim H^{1}(\Omega),$$ why? The authors offered a reference but it is French, but I don't understand French. the reference is "Grisvard,P. Characterisation de Quelques Esoaces d'interpolation. Archives Rational Mechanics and Analysis 1967,26,40-63"

Any comments and hints are welcome, thank you very much!!!

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monotone operator
monotone operator
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Elliptic partial differential equations with robin boundary condition and domain of fractional power of Robin Laplacian operator

This question have been posted in stack exchange but nobody care, and so I put it here. When I read the paper "On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation" by Igor Chueshov,Matthias Eller and Irena Lasiecka,I encounter a difficulty: Suppose that $\Omega$ is a simple connected domain with smooth boundary, the authors introduced a Robin Laplacian operator $$\Delta_{R}:L^{2}(\Omega)\to L^{2}(\Omega).$$This is an unbounded operator with the domain $$D(\Delta_{R})=\{u\in H^{2}(\Omega):\partial_{\nu}u+u=0\ on\ \partial\Omega\}.$$Moreover, the Robin Laplacian can be extended to a continuous operator $\Delta_{R}:H^{1}(\Omega)\to H^{1}(\Omega)'$ by $$(-\Delta_{R}u,v)_{L^{2}(\Omega)}=(\nabla u,\nabla v)_{L^{2}(\Omega)}+<u,v>_{L^{2}(\partial\Omega)}.$$ Then the authors say "this extension is the duality map $H^{1}(\Omega)$ into $(H^{1}(\Omega))'$" when we equip $H^{1}(\Omega)$ the norm $$\|u\|=\sqrt{(\nabla u,\nabla v)_{L^{2}(\Omega)}+<u,v>_{L^{2}(\partial\Omega)}}.$$ So, firstly, I want to know if this norm is equivalent to the usual Sobolev norm $$\|u\|=\sqrt{(\nabla u,\nabla v)_{L^{2}(\Omega)}+(u,v)_{L^{2}(\Omega)}}$$ when $u$ satisfies the robin boundary conditon? Next, the authors said $$D((-\Delta_{R}))^{\frac{1}{2}}\sim H^{1}(\Omega),$$ why? the authors offered an reference but it is french, but I don't understand french. the reference is "Grisvard,P. Characterisation de Quelques Esoaces d'interpolation. Archives Rational Mechanics and Analysis 1967,26,40-63"

Any comments and hints are welcome, thank you very much!!!