Abstract
Abstract We investigate the different notions of solutions to the double-phase equation - div ( | D u | p - 2 D u + a ( x ) | D u | q - 2 D u ) = 0 , -{\operatorname{div}(\lvert Du\rvert^{p-2}Du+a(x)\lvert Du\rvert^{q-2}Du)}=0, which is characterized by the fact that both ellipticity and growth switch between two different types of polynomial according to the position. We introduce the 𝒜 H ( ⋅ ) {\mathcal{A}_{H(\,{\cdot}\,)}} -harmonic functions of nonlinear potential theory and then show that 𝒜 H ( ⋅ ) {\mathcal{A}_{H(\,{\cdot}\,)}} -harmonic functions coincide with the distributional and viscosity solutions, respectively. This implies that the distributional and viscosity solutions are exactly the same.
Highlights
Let Ω be a bounded domain in Rn, n ≥ 2
It is worth to mention that a shorter proof for the equivalence of distributional and viscosity solutions for the p-Laplace equation was recently given in [21] by virtue of a technical regularization procedure via infimal convolutions
We prove the equivalence of two notions of solutions of (1.1) through justifying the distributional and viscosity solutions coincide with AH(·)-harmonic functions, respectively
Summary
Let Ω be a bounded domain in Rn, n ≥ 2. We are concerned with the relationship of distributional and viscosity solutions to the following double-phase problem. Equivalence; Double-phase equation; Viscosity solution; Distributional solution; AH(·)-harmonic function; Comparison principle. There are few results concerning the viscosity solutions for such kind of equations To this end, our interest in this work focuses on the different notions of solutions to Eq (1.1). It is worth to mention that a shorter proof for the equivalence of distributional and viscosity solutions for the p-Laplace equation was recently given in [21] by virtue of a technical regularization procedure via infimal convolutions. We prove the equivalence of two notions of solutions of (1.1) through justifying the distributional and viscosity solutions coincide with AH(·)-harmonic functions, respectively. To the best of our knowledge, the result that distributional solutions and AH(·)-harmonic functions of double-phase equation coincide is new.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
