We prove the local Hölder continuity for the weak solutions of the following elliptic partial differential equations of second order: <Equation ID="Equ52"> <EquationSource Format="TEX">$\begin{aligned} -\text {div} \, ( A(x) \nabla u(x) ) + {\textbf {b}}(x) \cdot \nabla u(x) + V(x) u(x) = 0 \qquad \text {in} \,\, \Omega , \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>-</mo> <mtext>div</mtext> <mspace width="0.166667em" /> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi mathvariant="bold">b</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>·</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="2em" /> <mtext>in</mtext> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is an open bounded subset of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$N \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, A(x) is an elliptic symmetric matrix with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^{\infty }(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-coefficients, and the lower-order terms <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\textbf {b}}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">b</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and V(x) are respectively a vector field and a function such that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$|{\textbf {b}}|^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="bold">b</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and V belonging to the Morrey spaces <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L^{1,\lambda }(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>λ</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$N-2<\lambda <N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>-</mo> <mn>2</mn> <mo><</mo> <mi>λ</mi> <mo><</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation>.

Hölder continuity for solutions of elliptic partial differential equations of second order

期刊论文
发表日期：

2026年1月28日

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Nicky K. Tumalun

摘要

We prove the local Hölder continuity for the weak solutions of the following elliptic partial differential equations of second order: $\begin{aligned} -\text {div} \, ( A(x) \nabla u(x) ) + {\textbf {b}}(x) \cdot \nabla u(x) + V(x) u(x) = 0 \qquad \text {in} \,\, \Omega , \end{aligned}$ $\begin{matrix} - div (A (x) \nabla u (x)) + b (x) \cdot \nabla u (x) + V (x) u (x) = 0 in Ω, \end{matrix}$ where $\Omega $ $Ω$ is an open bounded subset of $\mathbb {R}^N$ $R^{N}$ , $N \ge 3$ $N \geq 3$ , A(x) is an elliptic symmetric matrix with $L^{\infty }(\Omega )$ $L^{\infty} (Ω)$ -coefficients, and the lower-order terms ${\textbf {b}}(x)$ $b (x)$ and V(x) are respectively a vector field and a function such that $|{\textbf {b}}|^2$ ${| b |}^{2}$ and V belonging to the Morrey spaces $L^{1,\lambda }(\Omega )$ $L^{1, λ} (Ω)$ , for $N-2<\lambda <N$ $N - 2 < λ < N$ .