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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.

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We consider the fully nonlinear equation with variable-exponent double phase type degeneraciesUnder some appropriate assumptions, by making use of geometric tangential methods and combing a refined improvement-of-flatness approach with compactness and scaling techniques, we obtain the sharp local C 1,α regularity of viscosity solutions to such equations.

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