The main result of the paper is the Aleksandrov–Bakelman–Pucci–Krylov–Tso (ABPKT) maximum principle for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>-viscosity sub/super-solutions of fully nonlinear uniformly parabolic equations <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$u_t+F(t,x,u,Du,D^2u)=f(t,x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>+</mo> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>D</mi> <mi>u</mi> <mo>,</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(0,T]\times \Omega ,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> <mo>×</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Omega \subset {\mathbb {R}}^n.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> In this version of the maximum principle, the <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$L^{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> norm of f is taken over the so-called contact set. Equations have measurable and unbounded terms and we assume that the “drift” term which governs the dependence of F on the gradient variable is unbounded and is a function in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$L^{n+2}(Q).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Other versions of the ABPKT maximum principle for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-viscosity solutions and its pointwise version are also obtained. We use the maximum principles to prove various properties of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-viscosity solutions and build basic theory of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-viscosity solutions for uniformly parabolic equations with the unbounded drift term.

Aleksandrov–Bakelman–Pucci–Krylov–Tso maximum principle for $L^p$-viscosity solutions of parabolic equations with drift term in $L^q$

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2026年4月17日

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Shigeaki Koike,
Andrzej Święch

摘要

The main result of the paper is the Aleksandrov–Bakelman–Pucci–Krylov–Tso (ABPKT) maximum principle for $L^{n+1}$ $L^{n + 1}$ -viscosity sub/super-solutions of fully nonlinear uniformly parabolic equations $u_t+F(t,x,u,Du,D^2u)=f(t,x)$ $u_{t} + F (t, x, u, D u, D^{2} u) = f (t, x)$ in $(0,T]\times \Omega ,$ $(0, T] \times Ω,$ where $\Omega \subset {\mathbb {R}}^n.$ $Ω \subset R^{n} .$ In this version of the maximum principle, the $L^{n+1}$ $L^{n + 1}$ norm of f is taken over the so-called contact set. Equations have measurable and unbounded terms and we assume that the “drift” term which governs the dependence of F on the gradient variable is unbounded and is a function in $L^{n+2}(Q).$ $L^{n + 2} (Q) .$ Other versions of the ABPKT maximum principle for $L^p$ $L^{p}$ -viscosity solutions and its pointwise version are also obtained. We use the maximum principles to prove various properties of $L^p$ $L^{p}$ -viscosity solutions and build basic theory of $L^p$ $L^{p}$ -viscosity solutions for uniformly parabolic equations with the unbounded drift term.

Aleksandrov–Bakelman–Pucci–Krylov–Tso maximum principle for \(L^p\)-viscosity solutions of parabolic equations with drift term in \(L^q\)

摘要