Prove that a given function is harmonic

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Prove that the given function is harmonic in a region $Omega$: $$u(a)=frac1pi r^2 iint_D(a,r)u(x,y),dx,dy$$
if $barD(a,r)subsetOmega$ and $u$ is locally in $L_1.$




harmonic-functions




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  • See the section titled "The Mean Value Property" in Rudin's RCA.
    – Kavi Rama Murthy
    2 days ago










  • The problem is taken from there, but I have no idea how to solve it. Any hints?
    – Emin
    2 days ago










  • The integral equation you are asking to prove is called the mean value property of harmonic fictions: it is a standard property of harmonic functions and yup can find a proof of it in every textbook on potential theory, like the ones of Helms. It is possible also to define a generalized laplacian by using it as a brick, as Nikolai Gyunther did in the thirties of the twentieth century.
    – Daniele Tampieri
    2 days ago






  • 1




    My copy of Rudin's RCA states and proves this as a theorem when $u$ is continuous and the MVP is sated with integrals over circles instead of disks. First show that if $u$ is locally integrable and has the stated MVP then it is continuous. Now just adapt the proof in Rudin 's book using this new form of MVP.
    – Kavi Rama Murthy
    2 days ago














up vote
-2
down vote

favorite












Prove that the given function is harmonic in a region $Omega$: $$u(a)=frac1pi r^2 iint_D(a,r)u(x,y),dx,dy$$
if $barD(a,r)subsetOmega$ and $u$ is locally in $L_1.$




harmonic-functions




share|cite|improve this question





















  • See the section titled "The Mean Value Property" in Rudin's RCA.
    – Kavi Rama Murthy
    2 days ago










  • The problem is taken from there, but I have no idea how to solve it. Any hints?
    – Emin
    2 days ago










  • The integral equation you are asking to prove is called the mean value property of harmonic fictions: it is a standard property of harmonic functions and yup can find a proof of it in every textbook on potential theory, like the ones of Helms. It is possible also to define a generalized laplacian by using it as a brick, as Nikolai Gyunther did in the thirties of the twentieth century.
    – Daniele Tampieri
    2 days ago






  • 1




    My copy of Rudin's RCA states and proves this as a theorem when $u$ is continuous and the MVP is sated with integrals over circles instead of disks. First show that if $u$ is locally integrable and has the stated MVP then it is continuous. Now just adapt the proof in Rudin 's book using this new form of MVP.
    – Kavi Rama Murthy
    2 days ago












up vote
-2
down vote

favorite









up vote
-2
down vote

favorite











Prove that the given function is harmonic in a region $Omega$: $$u(a)=frac1pi r^2 iint_D(a,r)u(x,y),dx,dy$$
if $barD(a,r)subsetOmega$ and $u$ is locally in $L_1.$




harmonic-functions




share|cite|improve this question













Prove that the given function is harmonic in a region $Omega$: $$u(a)=frac1pi r^2 iint_D(a,r)u(x,y),dx,dy$$
if $barD(a,r)subsetOmega$ and $u$ is locally in $L_1.$





harmonic-functions





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited 2 days ago









Michael Hardy

204k23185460




204k23185460









asked 2 days ago









Emin

1,27621330




1,27621330











  • See the section titled "The Mean Value Property" in Rudin's RCA.
    – Kavi Rama Murthy
    2 days ago










  • The problem is taken from there, but I have no idea how to solve it. Any hints?
    – Emin
    2 days ago










  • The integral equation you are asking to prove is called the mean value property of harmonic fictions: it is a standard property of harmonic functions and yup can find a proof of it in every textbook on potential theory, like the ones of Helms. It is possible also to define a generalized laplacian by using it as a brick, as Nikolai Gyunther did in the thirties of the twentieth century.
    – Daniele Tampieri
    2 days ago






  • 1




    My copy of Rudin's RCA states and proves this as a theorem when $u$ is continuous and the MVP is sated with integrals over circles instead of disks. First show that if $u$ is locally integrable and has the stated MVP then it is continuous. Now just adapt the proof in Rudin 's book using this new form of MVP.
    – Kavi Rama Murthy
    2 days ago
















  • See the section titled "The Mean Value Property" in Rudin's RCA.
    – Kavi Rama Murthy
    2 days ago










  • The problem is taken from there, but I have no idea how to solve it. Any hints?
    – Emin
    2 days ago










  • The integral equation you are asking to prove is called the mean value property of harmonic fictions: it is a standard property of harmonic functions and yup can find a proof of it in every textbook on potential theory, like the ones of Helms. It is possible also to define a generalized laplacian by using it as a brick, as Nikolai Gyunther did in the thirties of the twentieth century.
    – Daniele Tampieri
    2 days ago






  • 1




    My copy of Rudin's RCA states and proves this as a theorem when $u$ is continuous and the MVP is sated with integrals over circles instead of disks. First show that if $u$ is locally integrable and has the stated MVP then it is continuous. Now just adapt the proof in Rudin 's book using this new form of MVP.
    – Kavi Rama Murthy
    2 days ago















See the section titled "The Mean Value Property" in Rudin's RCA.
– Kavi Rama Murthy
2 days ago




See the section titled "The Mean Value Property" in Rudin's RCA.
– Kavi Rama Murthy
2 days ago












The problem is taken from there, but I have no idea how to solve it. Any hints?
– Emin
2 days ago




The problem is taken from there, but I have no idea how to solve it. Any hints?
– Emin
2 days ago












The integral equation you are asking to prove is called the mean value property of harmonic fictions: it is a standard property of harmonic functions and yup can find a proof of it in every textbook on potential theory, like the ones of Helms. It is possible also to define a generalized laplacian by using it as a brick, as Nikolai Gyunther did in the thirties of the twentieth century.
– Daniele Tampieri
2 days ago




The integral equation you are asking to prove is called the mean value property of harmonic fictions: it is a standard property of harmonic functions and yup can find a proof of it in every textbook on potential theory, like the ones of Helms. It is possible also to define a generalized laplacian by using it as a brick, as Nikolai Gyunther did in the thirties of the twentieth century.
– Daniele Tampieri
2 days ago




1




1




My copy of Rudin's RCA states and proves this as a theorem when $u$ is continuous and the MVP is sated with integrals over circles instead of disks. First show that if $u$ is locally integrable and has the stated MVP then it is continuous. Now just adapt the proof in Rudin 's book using this new form of MVP.
– Kavi Rama Murthy
2 days ago




My copy of Rudin's RCA states and proves this as a theorem when $u$ is continuous and the MVP is sated with integrals over circles instead of disks. First show that if $u$ is locally integrable and has the stated MVP then it is continuous. Now just adapt the proof in Rudin 's book using this new form of MVP.
– Kavi Rama Murthy
2 days ago















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