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• W4379536555 abstract "Abstract In this paper, we prove gradient continuity estimates for viscosity solutions to $$Delta _{p}^N u - u_t= f$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msubsup> <mml:mi>Δ</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msubsup> <mml:mi>u</mml:mi> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> in terms of the scaling critical $$L(n+2,1 )$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> norm of f , where $$Delta _{p}^N$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msubsup> <mml:mi>Δ</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msubsup> </mml:math> is the game theoretic normalized $$p-$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> </mml:mrow> </mml:math> Laplacian operator defined in (1.2) below. Our main result, Theorem 2.5 constitutes borderline gradient continuity estimate for u in terms of the modified parabolic Riesz potential $$textbf{P}^{f}_{n+1}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msubsup> <mml:mi>P</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>f</mml:mi> </mml:msubsup> </mml:math> as defined in (2.9) below. Moreover, for $$f in L^{m}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> </mml:math> with $$m>n+2$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>></mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , we also obtain Hölder continuity of the spatial gradient of the solution u , see Theorem 2.6 below. This improves the gradient Hölder continuity result in Attouchi and Parviainen (Commun Contemp Math 20(4):1750035, 2018) which considers bounded f . Our main results Theorem 2.5 and Theorem 2.6 are parabolic analogues of those in Banerjee and Munive (Commun Contemp Math 22(8):1950069, 2020). Moreover differently from that in Attouchi and Parviainen (Commun Contemp Math 20(4):1750035, 2018), our approach is independent of the Ishii–Lions method which is crucially used in Attouchi and Parviainen (Commun Contemp Math 20(4):1750035, 2018) to obtain Lipschitz estimates for homogeneous perturbed equations as an intermediate step." @default.
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• W4379536555 date "2023-06-06" @default.
• W4379536555 modified "2023-10-17" @default.
• W4379536555 title "Borderline Gradient Continuity for the Normalized p-Parabolic Operator" @default.
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