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Authorea

10.22541/au.169648215.54013495/v1

<article class="ltx_document"> <div id="p1" class="ltx_para"> documentclassarticle </div> <section id="S1" class="ltx_section"> <h2 class="ltx_title ltx_title_section"><span class="ltx_tag ltx_tag_section">1 </span>Abstract</h2> <div id="S1.p1" class="ltx_para"> We prove results on unique continuation at the boundary for the solutions of real analytic elliptic partial differential equations of the form <table id="S1.Ex1" class="ltx_equation ltx_eqn_table"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"/> <td class="ltx_eqn_cell ltx_align_center"><math id="S1.Ex1.m1" class="ltx_Math" alttext="\sum_{i,j=1}^{n}a_{ij}(x)\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}+%
\sum_{k=1}^{n}b_{k}(x)\frac{\partial u}{\partial x_{k}}+c(x)u=0" display="block"><mrow><mo movablelimits="false">∑</mo><msub><mi/><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn></mrow></msub><msup><mi/><mi>n</mi></msup><mi>a</mi><msub><mi/><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mrow><mo rspace="0em">∂</mo><msup><mi/><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo rspace="0em">∂</mo><mi>x</mi><msub><mi/><mi>i</mi></msub><mo lspace="0em" rspace="0em">∂</mo><mi>x</mi><msub><mi/><mi>j</mi></msub></mrow></mfrac><mo rspace="0.055em">+</mo><mo movablelimits="false">∑</mo><msub><mi/><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow></msub><msup><mi/><mi>n</mi></msup><mi>b</mi><msub><mi/><mi>k</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mrow><mo rspace="0em">∂</mo><mi>u</mi></mrow><mrow><mo rspace="0em">∂</mo><mi>x</mi><msub><mi/><mi>k</mi></msub></mrow></mfrac><mo>+</mo><mi>c</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>u</mi><mo>=</mo><mn>0</mn></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"/></tr></tbody> </table> This work is motivated by and generalized the main results of , <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">berhanu2021boundary</span>]</cite>,<cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">berhanu2021local</span>]</cite>, X.Huang et al in ,<cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">huang1993unique</span>]</cite>,<cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">huang1995hopf</span>]</cite> and M.S Baouendi and L.P. Rothschild in <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">baouendi1993local</span>]</cite> Key words: Elliptic partial differential equation ; Hopf Lemma ; Unique continuation principle ; Real analytic hypersurface ; Real analytic functions </div> </section> <section id="bib" class="ltx_bibliography"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul id="bib.L1" class="ltx_biblist"> </ul> </section>

0009-0009-4947-0218

Chengzhi

5 10 2023

This preprint is available at https://doi.org/10.22541/au.169648215.54013495/v1

documentclassarticle

1 Abstract

We prove results on unique continuation at the boundary for the solutions of real analytic elliptic partial differential equations of the form

\sum_{i,j=1}^{n}a_{ij}(x)\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}+% \sum_{k=1}^{n}b_{k}(x)\frac{\partial u}{\partial x_{k}}+c(x)u=0

This work is motivated by and generalized the main results of , [berhanu2021boundary],[berhanu2021local], X.Huang et al in ,[huang1993unique],[huang1995hopf] and M.S Baouendi and L.P. Rothschild in [baouendi1993local] Key words: Elliptic partial differential equation ; Hopf Lemma ; Unique continuation principle ; Real analytic hypersurface ; Real analytic functions

References

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