<abstract xmlns="http://www.w3.org/1999/xhtml">

Let Ω ⊂ ℝn<italic></italic> be an open set. We give a new characterization of zero trace functions <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2022-0020_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>f</mi><mo>∈</mo><mi>𝒞</mi><mrow><mo>(</mo><mover accent="true"><mi mathvariant="normal">Ω</mi><mo>¯</mo></mover><mo>)</mo></mrow><mo>∩</mo><msubsup><mrow><mi>W</mi></mrow><mn>0</mn><mrow><mn>1</mn><mo>,</mo><mi>p</mi><mrow><mo>(</mo><mo>.</mo><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><mi mathvariant="normal">Ω</mi><mo>)</mo></mrow></mrow></math>
<tex-math>f \in \mathcal{C}\left( {\bar \Omega } \right) \cap W_0^{1,p\left( . \right)}\left( \Omega \right)</tex-math>
</alternatives>
</inline-formula>. If in addition Ω is bounded, then we give a sufficient condition for which the mapping <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2022-0020_eq_002.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>f</mi><mo>↦</mo><msubsup><mrow><mi>ℒ</mi></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mo>.</mo><mo>)</mo></mrow><mo>,</mo><mi>f</mi></mrow><mi mathvariant="normal">Ω</mi></msubsup></mrow></math>
<tex-math>f \mapsto \mathcal{L}_{p\left( . \right),f}^\Omega</tex-math>
</alternatives>
</inline-formula> from a set of real extended functions <italic>f</italic> : <italic>∂</italic>Ω −→ ℝ to the nonlinear harmonic space (Ω,ℋℒ<italic>p</italic>(.)) is injective, where <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2022-0020_eq_003.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>ℒ</mi></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mo>.</mo><mo>)</mo></mrow><mo>,</mo><mi>f</mi></mrow><mi mathvariant="normal">Ω</mi></msubsup></mrow></math>
<tex-math>\mathcal{L}_{p\left( . \right),f}^\Omega</tex-math>
</alternatives>
</inline-formula> denotes the Perron-Wiener-Brelot solution for the Dirichlet problem:
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2022-0020_eq_004.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow><mo>{</mo> <mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mrow><mi>ℒ</mi></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mo>.</mo><mo>)</mo></mrow></mrow></msub><mi>u</mi><mo>:</mo><mo>=</mo><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mo>.</mo><mo>)</mo></mrow></mrow></msub><mi>u</mi><mo>+</mo><mi>ℬ</mi><mrow><mo>(</mo><mrow><mo>.</mo><mo>,</mo><mi>u</mi></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></mtd><mtd columnalign="left"><mrow><mi>i</mi><mi>n</mi><mi> </mi><mi mathvariant="normal">Ω</mi><mo>;</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>u</mi><mo>=</mo><mi>f</mi></mrow></mtd><mtd columnalign="left"><mrow><mi>o</mi><mi>n</mi><mi> </mi><mo>∂</mo><mi mathvariant="normal">Ω</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow> </mrow></mrow></math>
<tex-math>\left\{ {\matrix{{{\mathcal{L}_{p\left( . \right)}}u: = - {\Delta _{p\left( . \right)}}u + \mathcal{B}\left( {.,u} \right) = 0} \hfill &amp; {in\,\Omega ;} \hfill \cr {u = f} \hfill &amp; {on\,\partial \Omega ,} \hfill \cr } } \right.</tex-math>
</alternatives>
</disp-formula>
where ℬ is a given Carathéodory function satisfies some structural conditions.
</abstract>

Let Ω ⊂ ℝn be an open set. We give a new characterization of zero trace functions 


f∈𝒞(Ω¯)∩W01,p(.)(Ω)
f \in \mathcal{C}\left( {\bar \Omega } \right) \cap W_0^{1,p\left( . \right)}\left( \Omega  \right)

. If in addition Ω is bounded, then we give a sufficient condition for which the mapping 


f↦ℒp(.),fΩ
f \mapsto \mathcal{L}_{p\left( . \right),f}^\Omega

 from a set of real extended functions f : ∂Ω −→ ℝ to the nonlinear harmonic space (Ω,ℋℒp(.)) is injective, where 


ℒp(.),fΩ
\mathcal{L}_{p\left( . \right),f}^\Omega

 denotes the Perron-Wiener-Brelot solution for the Dirichlet problem:



{ ℒp(.)u:=-Δp(.)u+ℬ(.,u)=0in Ω;u=fon ∂Ω, 
\left\{ {\matrix{{{\mathcal{L}_{p\left( . \right)}}u: =  - {\Delta _{p\left( . \right)}}u + \mathcal{B}\left( {.,u} \right) = 0} \hfill & {in\,\Omega ;} \hfill  \cr {u = f} \hfill & {on\,\partial \Omega ,} \hfill  \cr  } } \right.


where ℬ is a given Carathéodory function satisfies some structural conditions.

Capacitary characterization of variable exponent Sobolev trace spaces

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Capacitary characterization of variable exponent Sobolev trace spaces"}

Let Ω ⊂ ℝn be an open set. We give a new characterization of zero trace functions 


f∈𝒞(Ω¯)∩W01,p(.)(Ω)
f \in \mathcal{C}\left( {\bar \Omega } \right) \cap...

Capacitary characterization of variable exponent Sobolev trace spaces

Published Online: Oct 11, 2022

Page range: 286 - 298

Received: May 24, 2022

Accepted: Sep 03, 2022

DOI: https://doi.org/10.2478/mjpaa-2022-0020

Keywords
Sobolev spaces with variable exponent, trace spaces, p(.)-capacities, nonlinear harmonic spaces

© 2022 Mohamed Berghout, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Capacitary characterization of variable exponent Sobolev trace spaces

Published Online: Oct 11, 2022

Page range: 286 - 298

Received: May 24, 2022

Accepted: Sep 03, 2022

DOI: https://doi.org/10.2478/mjpaa-2022-0020

KeywordsSobolev spaces with variable exponent, trace spaces, p(.)-capacities, nonlinear harmonic spaces

© 2022 Mohamed Berghout, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Sobolev spaces with variable exponent, trace spaces, p(.)-capacities, nonlinear harmonic spaces