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You can find a full proof (to my knowledge the simpler one currently known) in the paper [1] and in the book [2], chapter I, §2.1 pp. 14-21. The original proof of Arthur Korn is so long and involved that K.O. Friedrichs, who gave a much simpler yet sophisticated proof, had doubts on his validity: starting from the work of Friedrichs, several authors gave their (in general quite complex) proofs, until Olga Oleĭnik gave a much shorter and simpler one (despite being still not elementary).

New edit. While ordering my library, I noted reference [1b]: in this paper Oleĭnik an Kondratiev prove the classical second Korn inequality for bounded domains satisfying the cone condition (theorem 1,a three page proof) and for certain classes of unbounded domains. They also prove that the constant in the inequality are sharp in some precise sense.

References

[1] Vladimir Alexandrovitch Kondratiev, Olga Arsenievna Oleĭnik, "On Korn’s inequalities" (English), Comptes Rendus de l’Académie des Sciences, Série I, 308, No. 16, pp. 483-487 (1989), MR0995908, Zbl 0698.35067.

[1b] Vladimir Alexandrovitch Kondrat’ev, Olga Arsenievna Oleĭnik, "Hardy’s and Korn’s type inequalities and their applications". (English) Rendiconti di Matematica e delle sue Applicazioni, VII Serie 10, No. 3, 641-666 (1990), MR1080319, Zbl 0767.35020, also found in the commemorative book Scritti matematici. Dedicati a Maria Adelaide Sneider, Università "La Sapienza", 415-440 (1990).

[2] Olga Arsenievna Oleĭnik, Alexei Stanislavovich Shamaev, Grigorii Andronikovich Yosifian, Mathematical problems in elasticity and homogenization. (English) Studies in Mathematics and its Applications. 26. Amsterdam-London-New York-Tokyo: North- Holland, pp. xiii+398 (1992), ISBN: 0-444-88441-6, MR1195131, Zbl 0768.73003.

You can find a full proof (to my knowledge the simpler one currently known) in the paper [1] and in the book [2], chapter I, §2.1 pp. 14-21. The original proof of Arthur Korn is so long and involved that K.O. Friedrichs, who gave a much simpler yet sophisticated proof, had doubts on his validity: starting from the work of Friedrichs, several authors gave their (in general quite complex) proofs, until Olga Oleĭnik gave a much shorter and simpler one (despite being still not elementary).

New edit. While ordering my library, I noted reference [1b]: in this paper Oleĭnik an Kondratiev prove the classical second Korn inequality for bounded domains satisfying the cone condition (theorem 1, a three page proof) and for certain classes of unbounded domains. They also prove that the constants in the inequality are sharp in some precise sense.

References

[1] Vladimir Alexandrovitch Kondratiev, Olga Arsenievna Oleĭnik, "On Korn’s inequalities" (English), Comptes Rendus de l’Académie des Sciences, Série I, 308, No. 16, pp. 483-487 (1989), MR0995908, Zbl 0698.35067.

[1b] Vladimir Alexandrovitch Kondrat’ev, Olga Arsenievna Oleĭnik, "Hardy’s and Korn’s type inequalities and their applications". (English) Rendiconti di Matematica e delle sue Applicazioni, VII Serie 10, No. 3, 641-666 (1990), MR1080319, Zbl 0767.35020, also found in the commemorative book Scritti matematici. Dedicati a Maria Adelaide Sneider, Università "La Sapienza", 415-440 (1990).

[2] Olga Arsenievna Oleĭnik, Alexei Stanislavovich Shamaev, Grigorii Andronikovich Yosifian, Mathematical problems in elasticity and homogenization. (English) Studies in Mathematics and its Applications. 26. Amsterdam-London-New York-Tokyo: North- Holland, pp. xiii+398 (1992), ISBN: 0-444-88441-6, MR1195131, Zbl 0768.73003.

You can find a full proof (to my knowledge the simpler one currently known) in the paper [1] and in the book [2], chapter I, §2.1 pp. 14-21. The original proof of Arthur Korn is so long and involved that K.O. Friedrichs, who gave a much simpler yet sophisticated proof, had doubts on his validity: starting from the work of Friedrichs, several authors gave their (in general quite complex) proofs, until Olga Oleĭnik gave a much shorter and simpler one (despite being still not elementary).

References

[1] Vladimir Alexandrovitch Kondratiev, Olga Arsenievna Oleĭnik, "On Korn’s inequalities" (English), Comptes Rendus de l’Académie des Sciences, Série I, 308, No. 16, pp. 483-487 (1989), MR0995908, Zbl 0698.35067.

[2] Olga Arsenievna Oleĭnik, Alexei Stanislavovich Shamaev, Grigorii Andronikovich Yosifian, Mathematical problems in elasticity and homogenization. (English) Studies in Mathematics and its Applications. 26. Amsterdam-London-New York-Tokyo: North- Holland, pp. xiii+398 (1992), ISBN: 0-444-88441-6, MR1195131, Zbl 0768.73003.

You can find a full proof (to my knowledge the simpler one currently known) in the paper [1] and in the book [2], chapter I, §2.1 pp. 14-21. The original proof of Arthur Korn is so long and involved that K.O. Friedrichs, who gave a much simpler yet sophisticated proof, had doubts on his validity: starting from the work of Friedrichs, several authors gave their (in general quite complex) proofs, until Olga Oleĭnik gave a much shorter and simpler one (despite being still not elementary).

New edit. While ordering my library, I noted reference [1b]: in this paper Oleĭnik an Kondratiev prove the classical second Korn inequality for bounded domains satisfying the cone condition (theorem 1,a three page proof) and for certain classes of unbounded domains. They also prove that the constant in the inequality are sharp in some precise sense.

References

[1] Vladimir Alexandrovitch Kondratiev, Olga Arsenievna Oleĭnik, "On Korn’s inequalities" (English), Comptes Rendus de l’Académie des Sciences, Série I, 308, No. 16, pp. 483-487 (1989), MR0995908, Zbl 0698.35067.

[1b] Vladimir Alexandrovitch Kondrat’ev, Olga Arsenievna Oleĭnik, "Hardy’s and Korn’s type inequalities and their applications". (English) Rendiconti di Matematica e delle sue Applicazioni, VII Serie 10, No. 3, 641-666 (1990), MR1080319, Zbl 0767.35020, also found in the commemorative book Scritti matematici. Dedicati a Maria Adelaide Sneider, Università "La Sapienza", 415-440 (1990).

[2] Olga Arsenievna Oleĭnik, Alexei Stanislavovich Shamaev, Grigorii Andronikovich Yosifian, Mathematical problems in elasticity and homogenization. (English) Studies in Mathematics and its Applications. 26. Amsterdam-London-New York-Tokyo: North- Holland, pp. xiii+398 (1992), ISBN: 0-444-88441-6, MR1195131, Zbl 0768.73003.

You can find a full proof (to my knowledge the simpler one currently known) in the paper [1] and in the book [2], chapter I, §2.1 pp. 14-21. The original proof of Arthur Korn is so long and involved that K.O. Friedrichs, who gave a much simpler yet sophisticated proof, had doubts on his validity: starting from the work of Friedriks, several authors gave their (in general quite complex) proofs, until Olga Oleĭnik gave a much shorter and simpler one (despite being still not elementary).

References

[1] Vladimir Alexandrovitch Kondratiev, Olga Arsenievna Oleĭnik, "On Korn’s inequalities" (English), Comptes Rendus de l’Académie des Sciences, Série I, 308, No. 16, pp. 483-487 (1989), MR0995908, Zbl 0698.35067.

[2] Olga Arsenievna Oleĭnik, Alexei Stanislavovich Shamaev, Grigorii Andronikovich Yosifian, Mathematical problems in elasticity and homogenization. (English) Studies in Mathematics and its Applications. 26. Amsterdam-London-New York-Tokyo: North- Holland, pp. xiii+398 (1992), ISBN: 0-444-88441-6, MR1195131, Zbl 0768.73003.

You can find a full proof (to my knowledge the simpler one currently known) in the paper [1] and in the book [2], chapter I, §2.1 pp. 14-21. The original proof of Arthur Korn is so long and involved that K.O. Friedrichs, who gave a much simpler yet sophisticated proof, had doubts on his validity: starting from the work of Friedrichs, several authors gave their (in general quite complex) proofs, until Olga Oleĭnik gave a much shorter and simpler one (despite being still not elementary).

References

[1] Vladimir Alexandrovitch Kondratiev, Olga Arsenievna Oleĭnik, "On Korn’s inequalities" (English), Comptes Rendus de l’Académie des Sciences, Série I, 308, No. 16, pp. 483-487 (1989), MR0995908, Zbl 0698.35067.

[2] Olga Arsenievna Oleĭnik, Alexei Stanislavovich Shamaev, Grigorii Andronikovich Yosifian, Mathematical problems in elasticity and homogenization. (English) Studies in Mathematics and its Applications. 26. Amsterdam-London-New York-Tokyo: North- Holland, pp. xiii+398 (1992), ISBN: 0-444-88441-6, MR1195131, Zbl 0768.73003.